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G = C7×C23.37C23order 448 = 26·7

Direct product of C7 and C23.37C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C23.37C23
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C22⋊C4 — C7×C22⋊Q8 — C7×C23.37C23
 Lower central C1 — C22 — C7×C23.37C23
 Upper central C1 — C2×C28 — C7×C23.37C23

Generators and relations for C7×C23.37C23
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 274 in 222 conjugacy classes, 170 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×C28, C2×C28, C2×C28, C7×Q8, C22×C14, C23.37C23, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, Q8×C14, C2×C4×C28, C7×C42⋊C2, Q8×C28, C7×C22⋊Q8, C7×C42.C2, C7×C4⋊Q8, C7×C23.37C23
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C4○D4, C24, C2×C14, C22×Q8, C2×C4○D4, C7×Q8, C22×C14, C23.37C23, Q8×C14, C7×C4○D4, C23×C14, Q8×C2×C14, C14×C4○D4, C7×C23.37C23

Smallest permutation representation of C7×C23.37C23
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)
(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 160)(142 161)(143 155)(144 156)(145 157)(146 158)(147 159)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 188)(170 189)(171 183)(172 184)(173 185)(174 186)(175 187)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 216)(198 217)(199 211)(200 212)(201 213)(202 214)(203 215)
(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)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 132)(114 133)(115 127)(116 128)(117 129)(118 130)(119 131)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 160)(142 161)(143 155)(144 156)(145 157)(146 158)(147 159)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 188)(170 189)(171 183)(172 184)(173 185)(174 186)(175 187)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 216)(198 217)(199 211)(200 212)(201 213)(202 214)(203 215)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 103)(79 104)(80 105)(81 99)(82 100)(83 101)(84 102)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 131)(107 132)(108 133)(109 127)(110 128)(111 129)(112 130)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 159)(135 160)(136 161)(137 155)(138 156)(139 157)(140 158)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 187)(163 188)(164 189)(165 183)(166 184)(167 185)(168 186)(169 179)(170 180)(171 181)(172 182)(173 176)(174 177)(175 178)(190 215)(191 216)(192 217)(193 211)(194 212)(195 213)(196 214)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 147 35 150)(2 141 29 151)(3 142 30 152)(4 143 31 153)(5 144 32 154)(6 145 33 148)(7 146 34 149)(8 118 18 121)(9 119 19 122)(10 113 20 123)(11 114 21 124)(12 115 15 125)(13 116 16 126)(14 117 17 120)(22 129 223 111)(23 130 224 112)(24 131 218 106)(25 132 219 107)(26 133 220 108)(27 127 221 109)(28 128 222 110)(36 156 44 138)(37 157 45 139)(38 158 46 140)(39 159 47 134)(40 160 48 135)(41 161 49 136)(42 155 43 137)(50 187 75 162)(51 188 76 163)(52 189 77 164)(53 183 71 165)(54 184 72 166)(55 185 73 167)(56 186 74 168)(57 179 67 169)(58 180 68 170)(59 181 69 171)(60 182 70 172)(61 176 64 173)(62 177 65 174)(63 178 66 175)(78 215 103 190)(79 216 104 191)(80 217 105 192)(81 211 99 193)(82 212 100 194)(83 213 101 195)(84 214 102 196)(85 207 95 197)(86 208 96 198)(87 209 97 199)(88 210 98 200)(89 204 92 201)(90 205 93 202)(91 206 94 203)
(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)(8 177 18 174)(9 178 19 175)(10 179 20 169)(11 180 21 170)(12 181 15 171)(13 182 16 172)(14 176 17 173)(22 167 223 185)(23 168 224 186)(24 162 218 187)(25 163 219 188)(26 164 220 189)(27 165 221 183)(28 166 222 184)(36 100 44 82)(37 101 45 83)(38 102 46 84)(39 103 47 78)(40 104 48 79)(41 105 49 80)(42 99 43 81)(50 131 75 106)(51 132 76 107)(52 133 77 108)(53 127 71 109)(54 128 72 110)(55 129 73 111)(56 130 74 112)(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)(134 190 159 215)(135 191 160 216)(136 192 161 217)(137 193 155 211)(138 194 156 212)(139 195 157 213)(140 196 158 214)(141 197 151 207)(142 198 152 208)(143 199 153 209)(144 200 154 210)(145 201 148 204)(146 202 149 205)(147 203 150 206)
(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)(8 202 23 214)(9 203 24 215)(10 197 25 216)(11 198 26 217)(12 199 27 211)(13 200 28 212)(14 201 22 213)(15 209 221 193)(16 210 222 194)(17 204 223 195)(18 205 224 196)(19 206 218 190)(20 207 219 191)(21 208 220 192)(29 76 48 57)(30 77 49 58)(31 71 43 59)(32 72 44 60)(33 73 45 61)(34 74 46 62)(35 75 47 63)(78 122 94 106)(79 123 95 107)(80 124 96 108)(81 125 97 109)(82 126 98 110)(83 120 92 111)(84 121 93 112)(85 132 104 113)(86 133 105 114)(87 127 99 115)(88 128 100 116)(89 129 101 117)(90 130 102 118)(91 131 103 119)(134 178 150 162)(135 179 151 163)(136 180 152 164)(137 181 153 165)(138 182 154 166)(139 176 148 167)(140 177 149 168)(141 188 160 169)(142 189 161 170)(143 183 155 171)(144 184 156 172)(145 185 157 173)(146 186 158 174)(147 187 159 175)

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), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,216)(198,217)(199,211)(200,212)(201,213)(202,214)(203,215), (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)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,132)(114,133)(115,127)(116,128)(117,129)(118,130)(119,131)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,216)(198,217)(199,211)(200,212)(201,213)(202,214)(203,215), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,103)(79,104)(80,105)(81,99)(82,100)(83,101)(84,102)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,131)(107,132)(108,133)(109,127)(110,128)(111,129)(112,130)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,159)(135,160)(136,161)(137,155)(138,156)(139,157)(140,158)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,187)(163,188)(164,189)(165,183)(166,184)(167,185)(168,186)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,215)(191,216)(192,217)(193,211)(194,212)(195,213)(196,214)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,147,35,150)(2,141,29,151)(3,142,30,152)(4,143,31,153)(5,144,32,154)(6,145,33,148)(7,146,34,149)(8,118,18,121)(9,119,19,122)(10,113,20,123)(11,114,21,124)(12,115,15,125)(13,116,16,126)(14,117,17,120)(22,129,223,111)(23,130,224,112)(24,131,218,106)(25,132,219,107)(26,133,220,108)(27,127,221,109)(28,128,222,110)(36,156,44,138)(37,157,45,139)(38,158,46,140)(39,159,47,134)(40,160,48,135)(41,161,49,136)(42,155,43,137)(50,187,75,162)(51,188,76,163)(52,189,77,164)(53,183,71,165)(54,184,72,166)(55,185,73,167)(56,186,74,168)(57,179,67,169)(58,180,68,170)(59,181,69,171)(60,182,70,172)(61,176,64,173)(62,177,65,174)(63,178,66,175)(78,215,103,190)(79,216,104,191)(80,217,105,192)(81,211,99,193)(82,212,100,194)(83,213,101,195)(84,214,102,196)(85,207,95,197)(86,208,96,198)(87,209,97,199)(88,210,98,200)(89,204,92,201)(90,205,93,202)(91,206,94,203), (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)(8,177,18,174)(9,178,19,175)(10,179,20,169)(11,180,21,170)(12,181,15,171)(13,182,16,172)(14,176,17,173)(22,167,223,185)(23,168,224,186)(24,162,218,187)(25,163,219,188)(26,164,220,189)(27,165,221,183)(28,166,222,184)(36,100,44,82)(37,101,45,83)(38,102,46,84)(39,103,47,78)(40,104,48,79)(41,105,49,80)(42,99,43,81)(50,131,75,106)(51,132,76,107)(52,133,77,108)(53,127,71,109)(54,128,72,110)(55,129,73,111)(56,130,74,112)(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)(134,190,159,215)(135,191,160,216)(136,192,161,217)(137,193,155,211)(138,194,156,212)(139,195,157,213)(140,196,158,214)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (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)(8,202,23,214)(9,203,24,215)(10,197,25,216)(11,198,26,217)(12,199,27,211)(13,200,28,212)(14,201,22,213)(15,209,221,193)(16,210,222,194)(17,204,223,195)(18,205,224,196)(19,206,218,190)(20,207,219,191)(21,208,220,192)(29,76,48,57)(30,77,49,58)(31,71,43,59)(32,72,44,60)(33,73,45,61)(34,74,46,62)(35,75,47,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,132,104,113)(86,133,105,114)(87,127,99,115)(88,128,100,116)(89,129,101,117)(90,130,102,118)(91,131,103,119)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,188,160,169)(142,189,161,170)(143,183,155,171)(144,184,156,172)(145,185,157,173)(146,186,158,174)(147,187,159,175)>;

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), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,216)(198,217)(199,211)(200,212)(201,213)(202,214)(203,215), (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)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,132)(114,133)(115,127)(116,128)(117,129)(118,130)(119,131)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,216)(198,217)(199,211)(200,212)(201,213)(202,214)(203,215), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,44)(37,45)(38,46)(39,47)(40,48)(41,49)(42,43)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,103)(79,104)(80,105)(81,99)(82,100)(83,101)(84,102)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,131)(107,132)(108,133)(109,127)(110,128)(111,129)(112,130)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,159)(135,160)(136,161)(137,155)(138,156)(139,157)(140,158)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,187)(163,188)(164,189)(165,183)(166,184)(167,185)(168,186)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,215)(191,216)(192,217)(193,211)(194,212)(195,213)(196,214)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,147,35,150)(2,141,29,151)(3,142,30,152)(4,143,31,153)(5,144,32,154)(6,145,33,148)(7,146,34,149)(8,118,18,121)(9,119,19,122)(10,113,20,123)(11,114,21,124)(12,115,15,125)(13,116,16,126)(14,117,17,120)(22,129,223,111)(23,130,224,112)(24,131,218,106)(25,132,219,107)(26,133,220,108)(27,127,221,109)(28,128,222,110)(36,156,44,138)(37,157,45,139)(38,158,46,140)(39,159,47,134)(40,160,48,135)(41,161,49,136)(42,155,43,137)(50,187,75,162)(51,188,76,163)(52,189,77,164)(53,183,71,165)(54,184,72,166)(55,185,73,167)(56,186,74,168)(57,179,67,169)(58,180,68,170)(59,181,69,171)(60,182,70,172)(61,176,64,173)(62,177,65,174)(63,178,66,175)(78,215,103,190)(79,216,104,191)(80,217,105,192)(81,211,99,193)(82,212,100,194)(83,213,101,195)(84,214,102,196)(85,207,95,197)(86,208,96,198)(87,209,97,199)(88,210,98,200)(89,204,92,201)(90,205,93,202)(91,206,94,203), (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)(8,177,18,174)(9,178,19,175)(10,179,20,169)(11,180,21,170)(12,181,15,171)(13,182,16,172)(14,176,17,173)(22,167,223,185)(23,168,224,186)(24,162,218,187)(25,163,219,188)(26,164,220,189)(27,165,221,183)(28,166,222,184)(36,100,44,82)(37,101,45,83)(38,102,46,84)(39,103,47,78)(40,104,48,79)(41,105,49,80)(42,99,43,81)(50,131,75,106)(51,132,76,107)(52,133,77,108)(53,127,71,109)(54,128,72,110)(55,129,73,111)(56,130,74,112)(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)(134,190,159,215)(135,191,160,216)(136,192,161,217)(137,193,155,211)(138,194,156,212)(139,195,157,213)(140,196,158,214)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (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)(8,202,23,214)(9,203,24,215)(10,197,25,216)(11,198,26,217)(12,199,27,211)(13,200,28,212)(14,201,22,213)(15,209,221,193)(16,210,222,194)(17,204,223,195)(18,205,224,196)(19,206,218,190)(20,207,219,191)(21,208,220,192)(29,76,48,57)(30,77,49,58)(31,71,43,59)(32,72,44,60)(33,73,45,61)(34,74,46,62)(35,75,47,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,132,104,113)(86,133,105,114)(87,127,99,115)(88,128,100,116)(89,129,101,117)(90,130,102,118)(91,131,103,119)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,188,160,169)(142,189,161,170)(143,183,155,171)(144,184,156,172)(145,185,157,173)(146,186,158,174)(147,187,159,175) );

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)], [(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,160),(142,161),(143,155),(144,156),(145,157),(146,158),(147,159),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,188),(170,189),(171,183),(172,184),(173,185),(174,186),(175,187),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,216),(198,217),(199,211),(200,212),(201,213),(202,214),(203,215)], [(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),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103),(106,122),(107,123),(108,124),(109,125),(110,126),(111,120),(112,121),(113,132),(114,133),(115,127),(116,128),(117,129),(118,130),(119,131),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,160),(142,161),(143,155),(144,156),(145,157),(146,158),(147,159),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,188),(170,189),(171,183),(172,184),(173,185),(174,186),(175,187),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,216),(198,217),(199,211),(200,212),(201,213),(202,214),(203,215)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(36,44),(37,45),(38,46),(39,47),(40,48),(41,49),(42,43),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,103),(79,104),(80,105),(81,99),(82,100),(83,101),(84,102),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(106,131),(107,132),(108,133),(109,127),(110,128),(111,129),(112,130),(113,123),(114,124),(115,125),(116,126),(117,120),(118,121),(119,122),(134,159),(135,160),(136,161),(137,155),(138,156),(139,157),(140,158),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(162,187),(163,188),(164,189),(165,183),(166,184),(167,185),(168,186),(169,179),(170,180),(171,181),(172,182),(173,176),(174,177),(175,178),(190,215),(191,216),(192,217),(193,211),(194,212),(195,213),(196,214),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,147,35,150),(2,141,29,151),(3,142,30,152),(4,143,31,153),(5,144,32,154),(6,145,33,148),(7,146,34,149),(8,118,18,121),(9,119,19,122),(10,113,20,123),(11,114,21,124),(12,115,15,125),(13,116,16,126),(14,117,17,120),(22,129,223,111),(23,130,224,112),(24,131,218,106),(25,132,219,107),(26,133,220,108),(27,127,221,109),(28,128,222,110),(36,156,44,138),(37,157,45,139),(38,158,46,140),(39,159,47,134),(40,160,48,135),(41,161,49,136),(42,155,43,137),(50,187,75,162),(51,188,76,163),(52,189,77,164),(53,183,71,165),(54,184,72,166),(55,185,73,167),(56,186,74,168),(57,179,67,169),(58,180,68,170),(59,181,69,171),(60,182,70,172),(61,176,64,173),(62,177,65,174),(63,178,66,175),(78,215,103,190),(79,216,104,191),(80,217,105,192),(81,211,99,193),(82,212,100,194),(83,213,101,195),(84,214,102,196),(85,207,95,197),(86,208,96,198),(87,209,97,199),(88,210,98,200),(89,204,92,201),(90,205,93,202),(91,206,94,203)], [(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),(8,177,18,174),(9,178,19,175),(10,179,20,169),(11,180,21,170),(12,181,15,171),(13,182,16,172),(14,176,17,173),(22,167,223,185),(23,168,224,186),(24,162,218,187),(25,163,219,188),(26,164,220,189),(27,165,221,183),(28,166,222,184),(36,100,44,82),(37,101,45,83),(38,102,46,84),(39,103,47,78),(40,104,48,79),(41,105,49,80),(42,99,43,81),(50,131,75,106),(51,132,76,107),(52,133,77,108),(53,127,71,109),(54,128,72,110),(55,129,73,111),(56,130,74,112),(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),(134,190,159,215),(135,191,160,216),(136,192,161,217),(137,193,155,211),(138,194,156,212),(139,195,157,213),(140,196,158,214),(141,197,151,207),(142,198,152,208),(143,199,153,209),(144,200,154,210),(145,201,148,204),(146,202,149,205),(147,203,150,206)], [(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),(8,202,23,214),(9,203,24,215),(10,197,25,216),(11,198,26,217),(12,199,27,211),(13,200,28,212),(14,201,22,213),(15,209,221,193),(16,210,222,194),(17,204,223,195),(18,205,224,196),(19,206,218,190),(20,207,219,191),(21,208,220,192),(29,76,48,57),(30,77,49,58),(31,71,43,59),(32,72,44,60),(33,73,45,61),(34,74,46,62),(35,75,47,63),(78,122,94,106),(79,123,95,107),(80,124,96,108),(81,125,97,109),(82,126,98,110),(83,120,92,111),(84,121,93,112),(85,132,104,113),(86,133,105,114),(87,127,99,115),(88,128,100,116),(89,129,101,117),(90,130,102,118),(91,131,103,119),(134,178,150,162),(135,179,151,163),(136,180,152,164),(137,181,153,165),(138,182,154,166),(139,176,148,167),(140,177,149,168),(141,188,160,169),(142,189,161,170),(143,183,155,171),(144,184,156,172),(145,185,157,173),(146,186,158,174),(147,187,159,175)]])

196 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V 7A ··· 7F 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28CF 28CG ··· 28EB order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 Q8 C4○D4 C7×Q8 C7×C4○D4 kernel C7×C23.37C23 C2×C4×C28 C7×C42⋊C2 Q8×C28 C7×C22⋊Q8 C7×C42.C2 C7×C4⋊Q8 C23.37C23 C2×C42 C42⋊C2 C4×Q8 C22⋊Q8 C42.C2 C4⋊Q8 C2×C28 C28 C2×C4 C4 # reps 1 1 2 4 4 2 2 6 6 12 24 24 12 12 4 8 24 48

Matrix representation of C7×C23.37C23 in GL4(𝔽29) generated by

 20 0 0 0 0 20 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 28 0 0 0 0 1 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 28 0
,
 28 0 0 0 0 28 0 0 0 0 12 0 0 0 0 17
,
 17 0 0 0 0 17 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,1,0,0,1,0,0,0,0,0,0,28,0,0,1,0],[28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,17],[17,0,0,0,0,17,0,0,0,0,12,0,0,0,0,12] >;

C7×C23.37C23 in GAP, Magma, Sage, TeX

C_7\times C_2^3._{37}C_2^3
% in TeX

G:=Group("C7xC2^3.37C2^3");
// GroupNames label

G:=SmallGroup(448,1316);
// by ID

G=gap.SmallGroup(448,1316);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,4790,1192,416]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=f^2=d,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations

׿
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