direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23.4Q8, (C2×C28).312D4, C23.4(C7×Q8), (C22×C14).4Q8, C24.12(C2×C14), C22.75(D4×C14), C22.24(Q8×C14), C14.41(C4⋊1D4), C14.91(C22⋊Q8), (C23×C14).9C22, C2.C42⋊12C14, C23.82(C22×C14), (C22×C28).36C22, (C22×C14).463C23, C14.93(C22.D4), (C2×C4⋊C4)⋊8C14, (C14×C4⋊C4)⋊35C2, (C2×C4).19(C7×D4), C2.4(C7×C4⋊1D4), C2.10(C7×C22⋊Q8), (C2×C14).615(C2×D4), (C2×C14).112(C2×Q8), C22.42(C7×C4○D4), (C2×C22⋊C4).10C14, (C14×C22⋊C4).30C2, (C22×C4).23(C2×C14), (C2×C14).223(C4○D4), C2.9(C7×C22.D4), (C7×C2.C42)⋊28C2, SmallGroup(448,807)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.4Q8
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 330 in 186 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.4Q8, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C23×C14, C7×C2.C42, C14×C22⋊C4, C14×C4⋊C4, C7×C23.4Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C22⋊Q8, C22.D4, C4⋊1D4, C7×D4, C7×Q8, C22×C14, C23.4Q8, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊Q8, C7×C22.D4, C7×C4⋊1D4, C7×C23.4Q8
►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 41)(9 42)(10 36)(11 37)(12 38)(13 39)(14 40)(15 35)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(99 129)(100 130)(101 131)(102 132)(103 133)(104 127)(105 128)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 153)(121 154)(122 148)(123 149)(124 150)(125 151)(126 152)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)(211 220)(212 221)(213 222)(214 223)(215 224)(216 218)(217 219)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 217)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 69)(2 70)(3 64)(4 65)(5 66)(6 67)(7 68)(8 31)(9 32)(10 33)(11 34)(12 35)(13 29)(14 30)(15 38)(16 39)(17 40)(18 41)(19 42)(20 36)(21 37)(22 220)(23 221)(24 222)(25 223)(26 224)(27 218)(28 219)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(71 95)(72 96)(73 97)(74 98)(75 92)(76 93)(77 94)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(99 129)(100 130)(101 131)(102 132)(103 133)(104 127)(105 128)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 153)(121 154)(122 148)(123 149)(124 150)(125 151)(126 152)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 185)(156 186)(157 187)(158 188)(159 189)(160 183)(161 184)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 209)(177 210)(178 204)(179 205)(180 206)(181 207)(182 208)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)
(1 104 95 151)(2 105 96 152)(3 99 97 153)(4 100 98 154)(5 101 92 148)(6 102 93 149)(7 103 94 150)(8 155 22 209)(9 156 23 210)(10 157 24 204)(11 158 25 205)(12 159 26 206)(13 160 27 207)(14 161 28 208)(15 170 215 203)(16 171 216 197)(17 172 217 198)(18 173 211 199)(19 174 212 200)(20 175 213 201)(21 169 214 202)(29 183 218 181)(30 184 219 182)(31 185 220 176)(32 186 221 177)(33 187 222 178)(34 188 223 179)(35 189 224 180)(36 166 45 192)(37 167 46 193)(38 168 47 194)(39 162 48 195)(40 163 49 196)(41 164 43 190)(42 165 44 191)(50 139 83 106)(51 140 84 107)(52 134 78 108)(53 135 79 109)(54 136 80 110)(55 137 81 111)(56 138 82 112)(57 146 90 113)(58 147 91 114)(59 141 85 115)(60 142 86 116)(61 143 87 117)(62 144 88 118)(63 145 89 119)(64 129 73 120)(65 130 74 121)(66 131 75 122)(67 132 76 123)(68 133 77 124)(69 127 71 125)(70 128 72 126)
(1 171 59 207)(2 172 60 208)(3 173 61 209)(4 174 62 210)(5 175 63 204)(6 169 57 205)(7 170 58 206)(8 143 211 99)(9 144 212 100)(10 145 213 101)(11 146 214 102)(12 147 215 103)(13 141 216 104)(14 142 217 105)(15 150 26 114)(16 151 27 115)(17 152 28 116)(18 153 22 117)(19 154 23 118)(20 148 24 119)(21 149 25 113)(29 139 48 127)(30 140 49 128)(31 134 43 129)(32 135 44 130)(33 136 45 131)(34 137 46 132)(35 138 47 133)(36 122 222 110)(37 123 223 111)(38 124 224 112)(39 125 218 106)(40 126 219 107)(41 120 220 108)(42 121 221 109)(50 181 69 162)(51 182 70 163)(52 176 64 164)(53 177 65 165)(54 178 66 166)(55 179 67 167)(56 180 68 168)(71 195 83 183)(72 196 84 184)(73 190 78 185)(74 191 79 186)(75 192 80 187)(76 193 81 188)(77 194 82 189)(85 160 95 197)(86 161 96 198)(87 155 97 199)(88 156 98 200)(89 157 92 201)(90 158 93 202)(91 159 94 203)
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), (8,41)(9,42)(10,36)(11,37)(12,38)(13,39)(14,40)(15,35)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(99,129)(100,130)(101,131)(102,132)(103,133)(104,127)(105,128)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,153)(121,154)(122,148)(123,149)(124,150)(125,151)(126,152)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206)(211,220)(212,221)(213,222)(214,223)(215,224)(216,218)(217,219), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,69)(2,70)(3,64)(4,65)(5,66)(6,67)(7,68)(8,31)(9,32)(10,33)(11,34)(12,35)(13,29)(14,30)(15,38)(16,39)(17,40)(18,41)(19,42)(20,36)(21,37)(22,220)(23,221)(24,222)(25,223)(26,224)(27,218)(28,219)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(71,95)(72,96)(73,97)(74,98)(75,92)(76,93)(77,94)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,129)(100,130)(101,131)(102,132)(103,133)(104,127)(105,128)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,153)(121,154)(122,148)(123,149)(124,150)(125,151)(126,152)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,185)(156,186)(157,187)(158,188)(159,189)(160,183)(161,184)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,209)(177,210)(178,204)(179,205)(180,206)(181,207)(182,208)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198), (1,104,95,151)(2,105,96,152)(3,99,97,153)(4,100,98,154)(5,101,92,148)(6,102,93,149)(7,103,94,150)(8,155,22,209)(9,156,23,210)(10,157,24,204)(11,158,25,205)(12,159,26,206)(13,160,27,207)(14,161,28,208)(15,170,215,203)(16,171,216,197)(17,172,217,198)(18,173,211,199)(19,174,212,200)(20,175,213,201)(21,169,214,202)(29,183,218,181)(30,184,219,182)(31,185,220,176)(32,186,221,177)(33,187,222,178)(34,188,223,179)(35,189,224,180)(36,166,45,192)(37,167,46,193)(38,168,47,194)(39,162,48,195)(40,163,49,196)(41,164,43,190)(42,165,44,191)(50,139,83,106)(51,140,84,107)(52,134,78,108)(53,135,79,109)(54,136,80,110)(55,137,81,111)(56,138,82,112)(57,146,90,113)(58,147,91,114)(59,141,85,115)(60,142,86,116)(61,143,87,117)(62,144,88,118)(63,145,89,119)(64,129,73,120)(65,130,74,121)(66,131,75,122)(67,132,76,123)(68,133,77,124)(69,127,71,125)(70,128,72,126), (1,171,59,207)(2,172,60,208)(3,173,61,209)(4,174,62,210)(5,175,63,204)(6,169,57,205)(7,170,58,206)(8,143,211,99)(9,144,212,100)(10,145,213,101)(11,146,214,102)(12,147,215,103)(13,141,216,104)(14,142,217,105)(15,150,26,114)(16,151,27,115)(17,152,28,116)(18,153,22,117)(19,154,23,118)(20,148,24,119)(21,149,25,113)(29,139,48,127)(30,140,49,128)(31,134,43,129)(32,135,44,130)(33,136,45,131)(34,137,46,132)(35,138,47,133)(36,122,222,110)(37,123,223,111)(38,124,224,112)(39,125,218,106)(40,126,219,107)(41,120,220,108)(42,121,221,109)(50,181,69,162)(51,182,70,163)(52,176,64,164)(53,177,65,165)(54,178,66,166)(55,179,67,167)(56,180,68,168)(71,195,83,183)(72,196,84,184)(73,190,78,185)(74,191,79,186)(75,192,80,187)(76,193,81,188)(77,194,82,189)(85,160,95,197)(86,161,96,198)(87,155,97,199)(88,156,98,200)(89,157,92,201)(90,158,93,202)(91,159,94,203)>;
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), (8,41)(9,42)(10,36)(11,37)(12,38)(13,39)(14,40)(15,35)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(99,129)(100,130)(101,131)(102,132)(103,133)(104,127)(105,128)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,153)(121,154)(122,148)(123,149)(124,150)(125,151)(126,152)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206)(211,220)(212,221)(213,222)(214,223)(215,224)(216,218)(217,219), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,69)(2,70)(3,64)(4,65)(5,66)(6,67)(7,68)(8,31)(9,32)(10,33)(11,34)(12,35)(13,29)(14,30)(15,38)(16,39)(17,40)(18,41)(19,42)(20,36)(21,37)(22,220)(23,221)(24,222)(25,223)(26,224)(27,218)(28,219)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(71,95)(72,96)(73,97)(74,98)(75,92)(76,93)(77,94)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,129)(100,130)(101,131)(102,132)(103,133)(104,127)(105,128)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,153)(121,154)(122,148)(123,149)(124,150)(125,151)(126,152)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,185)(156,186)(157,187)(158,188)(159,189)(160,183)(161,184)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,209)(177,210)(178,204)(179,205)(180,206)(181,207)(182,208)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198), (1,104,95,151)(2,105,96,152)(3,99,97,153)(4,100,98,154)(5,101,92,148)(6,102,93,149)(7,103,94,150)(8,155,22,209)(9,156,23,210)(10,157,24,204)(11,158,25,205)(12,159,26,206)(13,160,27,207)(14,161,28,208)(15,170,215,203)(16,171,216,197)(17,172,217,198)(18,173,211,199)(19,174,212,200)(20,175,213,201)(21,169,214,202)(29,183,218,181)(30,184,219,182)(31,185,220,176)(32,186,221,177)(33,187,222,178)(34,188,223,179)(35,189,224,180)(36,166,45,192)(37,167,46,193)(38,168,47,194)(39,162,48,195)(40,163,49,196)(41,164,43,190)(42,165,44,191)(50,139,83,106)(51,140,84,107)(52,134,78,108)(53,135,79,109)(54,136,80,110)(55,137,81,111)(56,138,82,112)(57,146,90,113)(58,147,91,114)(59,141,85,115)(60,142,86,116)(61,143,87,117)(62,144,88,118)(63,145,89,119)(64,129,73,120)(65,130,74,121)(66,131,75,122)(67,132,76,123)(68,133,77,124)(69,127,71,125)(70,128,72,126), (1,171,59,207)(2,172,60,208)(3,173,61,209)(4,174,62,210)(5,175,63,204)(6,169,57,205)(7,170,58,206)(8,143,211,99)(9,144,212,100)(10,145,213,101)(11,146,214,102)(12,147,215,103)(13,141,216,104)(14,142,217,105)(15,150,26,114)(16,151,27,115)(17,152,28,116)(18,153,22,117)(19,154,23,118)(20,148,24,119)(21,149,25,113)(29,139,48,127)(30,140,49,128)(31,134,43,129)(32,135,44,130)(33,136,45,131)(34,137,46,132)(35,138,47,133)(36,122,222,110)(37,123,223,111)(38,124,224,112)(39,125,218,106)(40,126,219,107)(41,120,220,108)(42,121,221,109)(50,181,69,162)(51,182,70,163)(52,176,64,164)(53,177,65,165)(54,178,66,166)(55,179,67,167)(56,180,68,168)(71,195,83,183)(72,196,84,184)(73,190,78,185)(74,191,79,186)(75,192,80,187)(76,193,81,188)(77,194,82,189)(85,160,95,197)(86,161,96,198)(87,155,97,199)(88,156,98,200)(89,157,92,201)(90,158,93,202)(91,159,94,203) );
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)], [(8,41),(9,42),(10,36),(11,37),(12,38),(13,39),(14,40),(15,35),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(99,129),(100,130),(101,131),(102,132),(103,133),(104,127),(105,128),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,153),(121,154),(122,148),(123,149),(124,150),(125,151),(126,152),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206),(211,220),(212,221),(213,222),(214,223),(215,224),(216,218),(217,219)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,211),(23,212),(24,213),(25,214),(26,215),(27,216),(28,217),(29,39),(30,40),(31,41),(32,42),(33,36),(34,37),(35,38),(43,220),(44,221),(45,222),(46,223),(47,224),(48,218),(49,219),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,92),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(99,117),(100,118),(101,119),(102,113),(103,114),(104,115),(105,116),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,69),(2,70),(3,64),(4,65),(5,66),(6,67),(7,68),(8,31),(9,32),(10,33),(11,34),(12,35),(13,29),(14,30),(15,38),(16,39),(17,40),(18,41),(19,42),(20,36),(21,37),(22,220),(23,221),(24,222),(25,223),(26,224),(27,218),(28,219),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(71,95),(72,96),(73,97),(74,98),(75,92),(76,93),(77,94),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(99,129),(100,130),(101,131),(102,132),(103,133),(104,127),(105,128),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,153),(121,154),(122,148),(123,149),(124,150),(125,151),(126,152),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(155,185),(156,186),(157,187),(158,188),(159,189),(160,183),(161,184),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(176,209),(177,210),(178,204),(179,205),(180,206),(181,207),(182,208),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198)], [(1,104,95,151),(2,105,96,152),(3,99,97,153),(4,100,98,154),(5,101,92,148),(6,102,93,149),(7,103,94,150),(8,155,22,209),(9,156,23,210),(10,157,24,204),(11,158,25,205),(12,159,26,206),(13,160,27,207),(14,161,28,208),(15,170,215,203),(16,171,216,197),(17,172,217,198),(18,173,211,199),(19,174,212,200),(20,175,213,201),(21,169,214,202),(29,183,218,181),(30,184,219,182),(31,185,220,176),(32,186,221,177),(33,187,222,178),(34,188,223,179),(35,189,224,180),(36,166,45,192),(37,167,46,193),(38,168,47,194),(39,162,48,195),(40,163,49,196),(41,164,43,190),(42,165,44,191),(50,139,83,106),(51,140,84,107),(52,134,78,108),(53,135,79,109),(54,136,80,110),(55,137,81,111),(56,138,82,112),(57,146,90,113),(58,147,91,114),(59,141,85,115),(60,142,86,116),(61,143,87,117),(62,144,88,118),(63,145,89,119),(64,129,73,120),(65,130,74,121),(66,131,75,122),(67,132,76,123),(68,133,77,124),(69,127,71,125),(70,128,72,126)], [(1,171,59,207),(2,172,60,208),(3,173,61,209),(4,174,62,210),(5,175,63,204),(6,169,57,205),(7,170,58,206),(8,143,211,99),(9,144,212,100),(10,145,213,101),(11,146,214,102),(12,147,215,103),(13,141,216,104),(14,142,217,105),(15,150,26,114),(16,151,27,115),(17,152,28,116),(18,153,22,117),(19,154,23,118),(20,148,24,119),(21,149,25,113),(29,139,48,127),(30,140,49,128),(31,134,43,129),(32,135,44,130),(33,136,45,131),(34,137,46,132),(35,138,47,133),(36,122,222,110),(37,123,223,111),(38,124,224,112),(39,125,218,106),(40,126,219,107),(41,120,220,108),(42,121,221,109),(50,181,69,162),(51,182,70,163),(52,176,64,164),(53,177,65,165),(54,178,66,166),(55,179,67,167),(56,180,68,168),(71,195,83,183),(72,196,84,184),(73,190,78,185),(74,191,79,186),(75,192,80,187),(76,193,81,188),(77,194,82,189),(85,160,95,197),(86,161,96,198),(87,155,97,199),(88,156,98,200),(89,157,92,201),(90,158,93,202),(91,159,94,203)]])
154 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BB | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | Q8 | C4○D4 | C7×D4 | C7×Q8 | C7×C4○D4 |
| kernel | C7×C23.4Q8 | C7×C2.C42 | C14×C22⋊C4 | C14×C4⋊C4 | C23.4Q8 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C2×C28 | C22×C14 | C2×C14 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 3 | 3 | 6 | 6 | 18 | 18 | 6 | 2 | 6 | 36 | 12 | 36 |
Matrix representation of C7×C23.4Q8 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 24 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 28 |
| 28 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 27 |
| 0 | 0 | 0 | 0 | 22 | 25 |
| 13 | 27 | 0 | 0 | 0 | 0 |
| 27 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 17 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 2 |
| 0 | 0 | 0 | 0 | 6 | 4 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,0,0,0,0,0,0,23,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,13,0,0,0,0,0,28,0,0,0,0,0,0,1,24,0,0,0,0,0,28,0,0,0,0,0,0,1,4,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[28,16,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,1,12,0,0,0,0,0,0,4,22,0,0,0,0,27,25],[13,27,0,0,0,0,27,16,0,0,0,0,0,0,28,0,0,0,0,0,17,1,0,0,0,0,0,0,25,6,0,0,0,0,2,4] >;
C7×C23.4Q8 in GAP, Magma, Sage, TeX
C_7\times C_2^3._4Q_8
% in TeX
G:=Group("C7xC2^3.4Q8"); // GroupNames label
G:=SmallGroup(448,807);
// by ID
G=gap.SmallGroup(448,807);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,1184,2438,2403]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations