direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.33C24, C14.1552+ 1+4, C14.1132- 1+4, (C4×D4)⋊11C14, (D4×C28)⋊40C2, C22⋊Q8⋊8C14, C4⋊D4.8C14, C42.C2⋊4C14, C42⋊2C2⋊2C14, C42.37(C2×C14), (C2×C28).668C23, (C4×C28).278C22, (C2×C14).359C24, C22.D4⋊5C14, C2.5(C7×2- 1+4), C2.7(C7×2+ 1+4), (D4×C14).217C22, C23.38(C22×C14), (C22×C28).38C22, C22.33(C23×C14), (C22×C14).94C23, (Q8×C14).181C22, (C14×C4⋊C4)⋊46C2, (C2×C4⋊C4)⋊19C14, C4⋊C4.68(C2×C14), C2.16(C14×C4○D4), (C7×C22⋊Q8)⋊35C2, C22.5(C7×C4○D4), (C2×D4).31(C2×C14), C14.235(C2×C4○D4), (C7×C4⋊D4).18C2, (C2×Q8).25(C2×C14), (C7×C42.C2)⋊21C2, (C7×C42⋊2C2)⋊13C2, (C2×C14).53(C4○D4), C22⋊C4.15(C2×C14), (C7×C4⋊C4).392C22, (C22×C4).59(C2×C14), (C2×C4).26(C22×C14), (C7×C22.D4)⋊24C2, (C7×C22⋊C4).85C22, SmallGroup(448,1322)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22.33C24
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=g2=1, e2=c, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 322 in 218 conjugacy classes, 146 normal (38 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22.33C24, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C14×C4⋊C4, D4×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C22⋊Q8, C7×C22.D4, C7×C42.C2, C7×C42⋊2C2, C7×C22.33C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C14, C22.33C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2+ 1+4, C7×2- 1+4, C7×C22.33C24
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 127)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 211)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(36 47)(37 48)(38 49)(39 43)(40 44)(41 45)(42 46)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(169 179)(170 180)(171 181)(172 182)(173 176)(174 177)(175 178)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 148)(7 149)(8 118)(9 119)(10 113)(11 114)(12 115)(13 116)(14 117)(15 122)(16 123)(17 124)(18 125)(19 126)(20 120)(21 121)(22 132)(23 133)(24 127)(25 128)(26 129)(27 130)(28 131)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 138)(37 139)(38 140)(39 134)(40 135)(41 136)(42 137)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(1 91 35 94)(2 85 29 95)(3 86 30 96)(4 87 31 97)(5 88 32 98)(6 89 33 92)(7 90 34 93)(8 189 21 168)(9 183 15 162)(10 184 16 163)(11 185 17 164)(12 186 18 165)(13 187 19 166)(14 188 20 167)(22 173 223 176)(23 174 224 177)(24 175 218 178)(25 169 219 179)(26 170 220 180)(27 171 221 181)(28 172 222 182)(36 103 47 82)(37 104 48 83)(38 105 49 84)(39 99 43 78)(40 100 44 79)(41 101 45 80)(42 102 46 81)(50 127 71 106)(51 128 72 107)(52 129 73 108)(53 130 74 109)(54 131 75 110)(55 132 76 111)(56 133 77 112)(57 123 67 113)(58 124 68 114)(59 125 69 115)(60 126 70 116)(61 120 64 117)(62 121 65 118)(63 122 66 119)(134 203 155 206)(135 197 156 207)(136 198 157 208)(137 199 158 209)(138 200 159 210)(139 201 160 204)(140 202 161 205)(141 191 151 212)(142 192 152 213)(143 193 153 214)(144 194 154 215)(145 195 148 216)(146 196 149 217)(147 190 150 211)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 196 23 205)(9 190 24 206)(10 191 25 207)(11 192 26 208)(12 193 27 209)(13 194 28 210)(14 195 22 204)(15 211 218 203)(16 212 219 197)(17 213 220 198)(18 214 221 199)(19 215 222 200)(20 216 223 201)(21 217 224 202)(29 72 44 57)(30 73 45 58)(31 74 46 59)(32 75 47 60)(33 76 48 61)(34 77 49 62)(35 71 43 63)(78 106 94 122)(79 107 95 123)(80 108 96 124)(81 109 97 125)(82 110 98 126)(83 111 92 120)(84 112 93 121)(85 113 100 128)(86 114 101 129)(87 115 102 130)(88 116 103 131)(89 117 104 132)(90 118 105 133)(91 119 99 127)(134 175 150 183)(135 169 151 184)(136 170 152 185)(137 171 153 186)(138 172 154 187)(139 173 148 188)(140 174 149 189)(141 163 156 179)(142 164 157 180)(143 165 158 181)(144 166 159 182)(145 167 160 176)(146 168 161 177)(147 162 155 178)
(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 211)
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,122)(16,123)(17,124)(18,125)(19,126)(20,120)(21,121)(22,132)(23,133)(24,127)(25,128)(26,129)(27,130)(28,131)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,189,21,168)(9,183,15,162)(10,184,16,163)(11,185,17,164)(12,186,18,165)(13,187,19,166)(14,188,20,167)(22,173,223,176)(23,174,224,177)(24,175,218,178)(25,169,219,179)(26,170,220,180)(27,171,221,181)(28,172,222,182)(36,103,47,82)(37,104,48,83)(38,105,49,84)(39,99,43,78)(40,100,44,79)(41,101,45,80)(42,102,46,81)(50,127,71,106)(51,128,72,107)(52,129,73,108)(53,130,74,109)(54,131,75,110)(55,132,76,111)(56,133,77,112)(57,123,67,113)(58,124,68,114)(59,125,69,115)(60,126,70,116)(61,120,64,117)(62,121,65,118)(63,122,66,119)(134,203,155,206)(135,197,156,207)(136,198,157,208)(137,199,158,209)(138,200,159,210)(139,201,160,204)(140,202,161,205)(141,191,151,212)(142,192,152,213)(143,193,153,214)(144,194,154,215)(145,195,148,216)(146,196,149,217)(147,190,150,211), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,196,23,205)(9,190,24,206)(10,191,25,207)(11,192,26,208)(12,193,27,209)(13,194,28,210)(14,195,22,204)(15,211,218,203)(16,212,219,197)(17,213,220,198)(18,214,221,199)(19,215,222,200)(20,216,223,201)(21,217,224,202)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,106,94,122)(79,107,95,123)(80,108,96,124)(81,109,97,125)(82,110,98,126)(83,111,92,120)(84,112,93,121)(85,113,100,128)(86,114,101,129)(87,115,102,130)(88,116,103,131)(89,117,104,132)(90,118,105,133)(91,119,99,127)(134,175,150,183)(135,169,151,184)(136,170,152,185)(137,171,153,186)(138,172,154,187)(139,173,148,188)(140,174,149,189)(141,163,156,179)(142,164,157,180)(143,165,158,181)(144,166,159,182)(145,167,160,176)(146,168,161,177)(147,162,155,178), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,122)(16,123)(17,124)(18,125)(19,126)(20,120)(21,121)(22,132)(23,133)(24,127)(25,128)(26,129)(27,130)(28,131)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,189,21,168)(9,183,15,162)(10,184,16,163)(11,185,17,164)(12,186,18,165)(13,187,19,166)(14,188,20,167)(22,173,223,176)(23,174,224,177)(24,175,218,178)(25,169,219,179)(26,170,220,180)(27,171,221,181)(28,172,222,182)(36,103,47,82)(37,104,48,83)(38,105,49,84)(39,99,43,78)(40,100,44,79)(41,101,45,80)(42,102,46,81)(50,127,71,106)(51,128,72,107)(52,129,73,108)(53,130,74,109)(54,131,75,110)(55,132,76,111)(56,133,77,112)(57,123,67,113)(58,124,68,114)(59,125,69,115)(60,126,70,116)(61,120,64,117)(62,121,65,118)(63,122,66,119)(134,203,155,206)(135,197,156,207)(136,198,157,208)(137,199,158,209)(138,200,159,210)(139,201,160,204)(140,202,161,205)(141,191,151,212)(142,192,152,213)(143,193,153,214)(144,194,154,215)(145,195,148,216)(146,196,149,217)(147,190,150,211), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,196,23,205)(9,190,24,206)(10,191,25,207)(11,192,26,208)(12,193,27,209)(13,194,28,210)(14,195,22,204)(15,211,218,203)(16,212,219,197)(17,213,220,198)(18,214,221,199)(19,215,222,200)(20,216,223,201)(21,217,224,202)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,106,94,122)(79,107,95,123)(80,108,96,124)(81,109,97,125)(82,110,98,126)(83,111,92,120)(84,112,93,121)(85,113,100,128)(86,114,101,129)(87,115,102,130)(88,116,103,131)(89,117,104,132)(90,118,105,133)(91,119,99,127)(134,175,150,183)(135,169,151,184)(136,170,152,185)(137,171,153,186)(138,172,154,187)(139,173,148,188)(140,174,149,189)(141,163,156,179)(142,164,157,180)(143,165,158,181)(144,166,159,182)(145,167,160,176)(146,168,161,177)(147,162,155,178), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99),(106,122),(107,123),(108,124),(109,125),(110,126),(111,120),(112,121),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,127),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,211)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(36,47),(37,48),(38,49),(39,43),(40,44),(41,45),(42,46),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,123),(114,124),(115,125),(116,126),(117,120),(118,121),(119,122),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189),(169,179),(170,180),(171,181),(172,182),(173,176),(174,177),(175,178),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,148),(7,149),(8,118),(9,119),(10,113),(11,114),(12,115),(13,116),(14,117),(15,122),(16,123),(17,124),(18,125),(19,126),(20,120),(21,121),(22,132),(23,133),(24,127),(25,128),(26,129),(27,130),(28,131),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,138),(37,139),(38,140),(39,134),(40,135),(41,136),(42,137),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,91,35,94),(2,85,29,95),(3,86,30,96),(4,87,31,97),(5,88,32,98),(6,89,33,92),(7,90,34,93),(8,189,21,168),(9,183,15,162),(10,184,16,163),(11,185,17,164),(12,186,18,165),(13,187,19,166),(14,188,20,167),(22,173,223,176),(23,174,224,177),(24,175,218,178),(25,169,219,179),(26,170,220,180),(27,171,221,181),(28,172,222,182),(36,103,47,82),(37,104,48,83),(38,105,49,84),(39,99,43,78),(40,100,44,79),(41,101,45,80),(42,102,46,81),(50,127,71,106),(51,128,72,107),(52,129,73,108),(53,130,74,109),(54,131,75,110),(55,132,76,111),(56,133,77,112),(57,123,67,113),(58,124,68,114),(59,125,69,115),(60,126,70,116),(61,120,64,117),(62,121,65,118),(63,122,66,119),(134,203,155,206),(135,197,156,207),(136,198,157,208),(137,199,158,209),(138,200,159,210),(139,201,160,204),(140,202,161,205),(141,191,151,212),(142,192,152,213),(143,193,153,214),(144,194,154,215),(145,195,148,216),(146,196,149,217),(147,190,150,211)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,196,23,205),(9,190,24,206),(10,191,25,207),(11,192,26,208),(12,193,27,209),(13,194,28,210),(14,195,22,204),(15,211,218,203),(16,212,219,197),(17,213,220,198),(18,214,221,199),(19,215,222,200),(20,216,223,201),(21,217,224,202),(29,72,44,57),(30,73,45,58),(31,74,46,59),(32,75,47,60),(33,76,48,61),(34,77,49,62),(35,71,43,63),(78,106,94,122),(79,107,95,123),(80,108,96,124),(81,109,97,125),(82,110,98,126),(83,111,92,120),(84,112,93,121),(85,113,100,128),(86,114,101,129),(87,115,102,130),(88,116,103,131),(89,117,104,132),(90,118,105,133),(91,119,99,127),(134,175,150,183),(135,169,151,184),(136,170,152,185),(137,171,153,186),(138,172,154,187),(139,173,148,188),(140,174,149,189),(141,163,156,179),(142,164,157,180),(143,165,158,181),(144,166,159,182),(145,167,160,176),(146,168,161,177),(147,162,155,178)], [(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,211)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28CF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C4○D4 | C7×C4○D4 | 2+ 1+4 | 2- 1+4 | C7×2+ 1+4 | C7×2- 1+4 |
| kernel | C7×C22.33C24 | C14×C4⋊C4 | D4×C28 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C22.D4 | C7×C42.C2 | C7×C42⋊2C2 | C22.33C24 | C2×C4⋊C4 | C4×D4 | C4⋊D4 | C22⋊Q8 | C22.D4 | C42.C2 | C42⋊2C2 | C2×C14 | C22 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 3 | 4 | 2 | 2 | 6 | 6 | 12 | 6 | 18 | 24 | 12 | 12 | 4 | 24 | 1 | 1 | 6 | 6 |
Matrix representation of C7×C22.33C24 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 0 | 0 | 0 |
| 0 | 0 | 0 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 0 | 0 | 24 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 1 | 8 |
| 0 | 0 | 0 | 10 | 0 | 6 |
| 0 | 0 | 1 | 20 | 0 | 7 |
| 0 | 0 | 0 | 27 | 0 | 19 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 9 | 21 | 23 |
| 0 | 0 | 9 | 6 | 26 | 5 |
| 0 | 0 | 0 | 0 | 0 | 21 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 9 |
| 0 | 0 | 28 | 0 | 19 | 22 |
| 0 | 0 | 0 | 0 | 9 | 17 |
| 0 | 0 | 0 | 0 | 2 | 20 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 17 |
| 0 | 0 | 0 | 1 | 0 | 10 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,17,10,20,27,0,0,1,0,0,0,0,0,8,6,7,19],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,23,9,0,0,0,0,9,6,0,0,0,0,21,26,0,18,0,0,23,5,21,0],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,12,19,9,2,0,0,9,22,17,20],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,17,10,0,28] >;
C7×C22.33C24 in GAP, Magma, Sage, TeX
C_7\times C_2^2._{33}C_2^4 % in TeX
G:=Group("C7xC2^2.33C2^4"); // GroupNames label
G:=SmallGroup(448,1322);
// by ID
G=gap.SmallGroup(448,1322);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,1227,604,3363]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=g^2=1,e^2=c,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations