metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C7×Q8)⋊13D4, Q8⋊5(C7⋊D4), C7⋊5(Q8⋊D4), (C22×Q8)⋊1D7, (C2×C14)⋊14SD16, (C2×C28).303D4, C28.210(C2×D4), C22⋊3(Q8⋊D7), C14.74C22≀C2, Q8⋊Dic7⋊38C2, C28⋊7D4.14C2, (C2×Q8).168D14, C14.80(C2×SD16), C28.55D4⋊17C2, (C2×C28).477C23, (C22×C14).200D4, (C22×C4).156D14, C2.8(C24⋊D7), C23.88(C7⋊D4), (C2×D28).132C22, C4⋊Dic7.187C22, (Q8×C14).203C22, C2.22(C28.C23), C14.102(C8.C22), (C22×C28).203C22, (Q8×C2×C14)⋊1C2, (C2×Q8⋊D7)⋊23C2, C4.60(C2×C7⋊D4), C2.17(C2×Q8⋊D7), (C2×C14).560(C2×D4), (C2×C4).86(C7⋊D4), (C2×C7⋊C8).175C22, (C2×C4).562(C22×D7), C22.220(C2×C7⋊D4), SmallGroup(448,761)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C7×Q8)⋊13D4
G = < a,b,c,d,e | a7=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=dbd-1=ebe=b-1, dcd-1=ece=b-1c, ede=d-1 >
Subgroups: 692 in 158 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C7⋊C8, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×C14, Q8⋊D4, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, Q8⋊D7, C2×D28, C2×C7⋊D4, C22×C28, C22×C28, Q8×C14, Q8×C14, C28.55D4, Q8⋊Dic7, C28⋊7D4, C2×Q8⋊D7, Q8×C2×C14, (C7×Q8)⋊13D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8.C22, C7⋊D4, C22×D7, Q8⋊D4, Q8⋊D7, C2×C7⋊D4, C2×Q8⋊D7, C28.C23, C24⋊D7, (C7×Q8)⋊13D4
►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 50 8 43)(2 51 9 44)(3 52 10 45)(4 53 11 46)(5 54 12 47)(6 55 13 48)(7 56 14 49)(15 36 22 29)(16 37 23 30)(17 38 24 31)(18 39 25 32)(19 40 26 33)(20 41 27 34)(21 42 28 35)(57 99 64 106)(58 100 65 107)(59 101 66 108)(60 102 67 109)(61 103 68 110)(62 104 69 111)(63 105 70 112)(71 85 78 92)(72 86 79 93)(73 87 80 94)(74 88 81 95)(75 89 82 96)(76 90 83 97)(77 91 84 98)(113 155 120 162)(114 156 121 163)(115 157 122 164)(116 158 123 165)(117 159 124 166)(118 160 125 167)(119 161 126 168)(127 141 134 148)(128 142 135 149)(129 143 136 150)(130 144 137 151)(131 145 138 152)(132 146 139 153)(133 147 140 154)(169 218 176 211)(170 219 177 212)(171 220 178 213)(172 221 179 214)(173 222 180 215)(174 223 181 216)(175 224 182 217)(183 204 190 197)(184 205 191 198)(185 206 192 199)(186 207 193 200)(187 208 194 201)(188 209 195 202)(189 210 196 203)
(1 64 8 57)(2 65 9 58)(3 66 10 59)(4 67 11 60)(5 68 12 61)(6 69 13 62)(7 70 14 63)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)(113 218 120 211)(114 219 121 212)(115 220 122 213)(116 221 123 214)(117 222 124 215)(118 223 125 216)(119 224 126 217)(127 204 134 197)(128 205 135 198)(129 206 136 199)(130 207 137 200)(131 208 138 201)(132 209 139 202)(133 210 140 203)(141 183 148 190)(142 184 149 191)(143 185 150 192)(144 186 151 193)(145 187 152 194)(146 188 153 195)(147 189 154 196)(155 169 162 176)(156 170 163 177)(157 171 164 178)(158 172 165 179)(159 173 166 180)(160 174 167 181)(161 175 168 182)
(1 132 15 118)(2 131 16 117)(3 130 17 116)(4 129 18 115)(5 128 19 114)(6 127 20 113)(7 133 21 119)(8 139 22 125)(9 138 23 124)(10 137 24 123)(11 136 25 122)(12 135 26 121)(13 134 27 120)(14 140 28 126)(29 160 43 146)(30 159 44 145)(31 158 45 144)(32 157 46 143)(33 156 47 142)(34 155 48 141)(35 161 49 147)(36 167 50 153)(37 166 51 152)(38 165 52 151)(39 164 53 150)(40 163 54 149)(41 162 55 148)(42 168 56 154)(57 188 71 174)(58 187 72 173)(59 186 73 172)(60 185 74 171)(61 184 75 170)(62 183 76 169)(63 189 77 175)(64 195 78 181)(65 194 79 180)(66 193 80 179)(67 192 81 178)(68 191 82 177)(69 190 83 176)(70 196 84 182)(85 216 99 202)(86 215 100 201)(87 214 101 200)(88 213 102 199)(89 212 103 198)(90 211 104 197)(91 217 105 203)(92 223 106 209)(93 222 107 208)(94 221 108 207)(95 220 109 206)(96 219 110 205)(97 218 111 204)(98 224 112 210)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 119)(8 125)(9 124)(10 123)(11 122)(12 121)(13 120)(14 126)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 133)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 140)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 147)(36 153)(37 152)(38 151)(39 150)(40 149)(41 148)(42 154)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(49 161)(50 167)(51 166)(52 165)(53 164)(54 163)(55 162)(56 168)(57 174)(58 173)(59 172)(60 171)(61 170)(62 169)(63 175)(64 181)(65 180)(66 179)(67 178)(68 177)(69 176)(70 182)(71 188)(72 187)(73 186)(74 185)(75 184)(76 183)(77 189)(78 195)(79 194)(80 193)(81 192)(82 191)(83 190)(84 196)(85 202)(86 201)(87 200)(88 199)(89 198)(90 197)(91 203)(92 209)(93 208)(94 207)(95 206)(96 205)(97 204)(98 210)(99 216)(100 215)(101 214)(102 213)(103 212)(104 211)(105 217)(106 223)(107 222)(108 221)(109 220)(110 219)(111 218)(112 224)
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), (1,50,8,43)(2,51,9,44)(3,52,10,45)(4,53,11,46)(5,54,12,47)(6,55,13,48)(7,56,14,49)(15,36,22,29)(16,37,23,30)(17,38,24,31)(18,39,25,32)(19,40,26,33)(20,41,27,34)(21,42,28,35)(57,99,64,106)(58,100,65,107)(59,101,66,108)(60,102,67,109)(61,103,68,110)(62,104,69,111)(63,105,70,112)(71,85,78,92)(72,86,79,93)(73,87,80,94)(74,88,81,95)(75,89,82,96)(76,90,83,97)(77,91,84,98)(113,155,120,162)(114,156,121,163)(115,157,122,164)(116,158,123,165)(117,159,124,166)(118,160,125,167)(119,161,126,168)(127,141,134,148)(128,142,135,149)(129,143,136,150)(130,144,137,151)(131,145,138,152)(132,146,139,153)(133,147,140,154)(169,218,176,211)(170,219,177,212)(171,220,178,213)(172,221,179,214)(173,222,180,215)(174,223,181,216)(175,224,182,217)(183,204,190,197)(184,205,191,198)(185,206,192,199)(186,207,193,200)(187,208,194,201)(188,209,195,202)(189,210,196,203), (1,64,8,57)(2,65,9,58)(3,66,10,59)(4,67,11,60)(5,68,12,61)(6,69,13,62)(7,70,14,63)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105)(113,218,120,211)(114,219,121,212)(115,220,122,213)(116,221,123,214)(117,222,124,215)(118,223,125,216)(119,224,126,217)(127,204,134,197)(128,205,135,198)(129,206,136,199)(130,207,137,200)(131,208,138,201)(132,209,139,202)(133,210,140,203)(141,183,148,190)(142,184,149,191)(143,185,150,192)(144,186,151,193)(145,187,152,194)(146,188,153,195)(147,189,154,196)(155,169,162,176)(156,170,163,177)(157,171,164,178)(158,172,165,179)(159,173,166,180)(160,174,167,181)(161,175,168,182), (1,132,15,118)(2,131,16,117)(3,130,17,116)(4,129,18,115)(5,128,19,114)(6,127,20,113)(7,133,21,119)(8,139,22,125)(9,138,23,124)(10,137,24,123)(11,136,25,122)(12,135,26,121)(13,134,27,120)(14,140,28,126)(29,160,43,146)(30,159,44,145)(31,158,45,144)(32,157,46,143)(33,156,47,142)(34,155,48,141)(35,161,49,147)(36,167,50,153)(37,166,51,152)(38,165,52,151)(39,164,53,150)(40,163,54,149)(41,162,55,148)(42,168,56,154)(57,188,71,174)(58,187,72,173)(59,186,73,172)(60,185,74,171)(61,184,75,170)(62,183,76,169)(63,189,77,175)(64,195,78,181)(65,194,79,180)(66,193,80,179)(67,192,81,178)(68,191,82,177)(69,190,83,176)(70,196,84,182)(85,216,99,202)(86,215,100,201)(87,214,101,200)(88,213,102,199)(89,212,103,198)(90,211,104,197)(91,217,105,203)(92,223,106,209)(93,222,107,208)(94,221,108,207)(95,220,109,206)(96,219,110,205)(97,218,111,204)(98,224,112,210), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,119)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,126)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,133)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,140)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,147)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,154)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,161)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,168)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,175)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,182)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,189)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,196)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,203)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,210)(99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,217)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,224)>;
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), (1,50,8,43)(2,51,9,44)(3,52,10,45)(4,53,11,46)(5,54,12,47)(6,55,13,48)(7,56,14,49)(15,36,22,29)(16,37,23,30)(17,38,24,31)(18,39,25,32)(19,40,26,33)(20,41,27,34)(21,42,28,35)(57,99,64,106)(58,100,65,107)(59,101,66,108)(60,102,67,109)(61,103,68,110)(62,104,69,111)(63,105,70,112)(71,85,78,92)(72,86,79,93)(73,87,80,94)(74,88,81,95)(75,89,82,96)(76,90,83,97)(77,91,84,98)(113,155,120,162)(114,156,121,163)(115,157,122,164)(116,158,123,165)(117,159,124,166)(118,160,125,167)(119,161,126,168)(127,141,134,148)(128,142,135,149)(129,143,136,150)(130,144,137,151)(131,145,138,152)(132,146,139,153)(133,147,140,154)(169,218,176,211)(170,219,177,212)(171,220,178,213)(172,221,179,214)(173,222,180,215)(174,223,181,216)(175,224,182,217)(183,204,190,197)(184,205,191,198)(185,206,192,199)(186,207,193,200)(187,208,194,201)(188,209,195,202)(189,210,196,203), (1,64,8,57)(2,65,9,58)(3,66,10,59)(4,67,11,60)(5,68,12,61)(6,69,13,62)(7,70,14,63)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105)(113,218,120,211)(114,219,121,212)(115,220,122,213)(116,221,123,214)(117,222,124,215)(118,223,125,216)(119,224,126,217)(127,204,134,197)(128,205,135,198)(129,206,136,199)(130,207,137,200)(131,208,138,201)(132,209,139,202)(133,210,140,203)(141,183,148,190)(142,184,149,191)(143,185,150,192)(144,186,151,193)(145,187,152,194)(146,188,153,195)(147,189,154,196)(155,169,162,176)(156,170,163,177)(157,171,164,178)(158,172,165,179)(159,173,166,180)(160,174,167,181)(161,175,168,182), (1,132,15,118)(2,131,16,117)(3,130,17,116)(4,129,18,115)(5,128,19,114)(6,127,20,113)(7,133,21,119)(8,139,22,125)(9,138,23,124)(10,137,24,123)(11,136,25,122)(12,135,26,121)(13,134,27,120)(14,140,28,126)(29,160,43,146)(30,159,44,145)(31,158,45,144)(32,157,46,143)(33,156,47,142)(34,155,48,141)(35,161,49,147)(36,167,50,153)(37,166,51,152)(38,165,52,151)(39,164,53,150)(40,163,54,149)(41,162,55,148)(42,168,56,154)(57,188,71,174)(58,187,72,173)(59,186,73,172)(60,185,74,171)(61,184,75,170)(62,183,76,169)(63,189,77,175)(64,195,78,181)(65,194,79,180)(66,193,80,179)(67,192,81,178)(68,191,82,177)(69,190,83,176)(70,196,84,182)(85,216,99,202)(86,215,100,201)(87,214,101,200)(88,213,102,199)(89,212,103,198)(90,211,104,197)(91,217,105,203)(92,223,106,209)(93,222,107,208)(94,221,108,207)(95,220,109,206)(96,219,110,205)(97,218,111,204)(98,224,112,210), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,119)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,126)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,133)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,140)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,147)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,154)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,161)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,168)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,175)(64,181)(65,180)(66,179)(67,178)(68,177)(69,176)(70,182)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,189)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,196)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,203)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,210)(99,216)(100,215)(101,214)(102,213)(103,212)(104,211)(105,217)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,224) );
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)], [(1,50,8,43),(2,51,9,44),(3,52,10,45),(4,53,11,46),(5,54,12,47),(6,55,13,48),(7,56,14,49),(15,36,22,29),(16,37,23,30),(17,38,24,31),(18,39,25,32),(19,40,26,33),(20,41,27,34),(21,42,28,35),(57,99,64,106),(58,100,65,107),(59,101,66,108),(60,102,67,109),(61,103,68,110),(62,104,69,111),(63,105,70,112),(71,85,78,92),(72,86,79,93),(73,87,80,94),(74,88,81,95),(75,89,82,96),(76,90,83,97),(77,91,84,98),(113,155,120,162),(114,156,121,163),(115,157,122,164),(116,158,123,165),(117,159,124,166),(118,160,125,167),(119,161,126,168),(127,141,134,148),(128,142,135,149),(129,143,136,150),(130,144,137,151),(131,145,138,152),(132,146,139,153),(133,147,140,154),(169,218,176,211),(170,219,177,212),(171,220,178,213),(172,221,179,214),(173,222,180,215),(174,223,181,216),(175,224,182,217),(183,204,190,197),(184,205,191,198),(185,206,192,199),(186,207,193,200),(187,208,194,201),(188,209,195,202),(189,210,196,203)], [(1,64,8,57),(2,65,9,58),(3,66,10,59),(4,67,11,60),(5,68,12,61),(6,69,13,62),(7,70,14,63),(15,78,22,71),(16,79,23,72),(17,80,24,73),(18,81,25,74),(19,82,26,75),(20,83,27,76),(21,84,28,77),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,106,50,99),(44,107,51,100),(45,108,52,101),(46,109,53,102),(47,110,54,103),(48,111,55,104),(49,112,56,105),(113,218,120,211),(114,219,121,212),(115,220,122,213),(116,221,123,214),(117,222,124,215),(118,223,125,216),(119,224,126,217),(127,204,134,197),(128,205,135,198),(129,206,136,199),(130,207,137,200),(131,208,138,201),(132,209,139,202),(133,210,140,203),(141,183,148,190),(142,184,149,191),(143,185,150,192),(144,186,151,193),(145,187,152,194),(146,188,153,195),(147,189,154,196),(155,169,162,176),(156,170,163,177),(157,171,164,178),(158,172,165,179),(159,173,166,180),(160,174,167,181),(161,175,168,182)], [(1,132,15,118),(2,131,16,117),(3,130,17,116),(4,129,18,115),(5,128,19,114),(6,127,20,113),(7,133,21,119),(8,139,22,125),(9,138,23,124),(10,137,24,123),(11,136,25,122),(12,135,26,121),(13,134,27,120),(14,140,28,126),(29,160,43,146),(30,159,44,145),(31,158,45,144),(32,157,46,143),(33,156,47,142),(34,155,48,141),(35,161,49,147),(36,167,50,153),(37,166,51,152),(38,165,52,151),(39,164,53,150),(40,163,54,149),(41,162,55,148),(42,168,56,154),(57,188,71,174),(58,187,72,173),(59,186,73,172),(60,185,74,171),(61,184,75,170),(62,183,76,169),(63,189,77,175),(64,195,78,181),(65,194,79,180),(66,193,80,179),(67,192,81,178),(68,191,82,177),(69,190,83,176),(70,196,84,182),(85,216,99,202),(86,215,100,201),(87,214,101,200),(88,213,102,199),(89,212,103,198),(90,211,104,197),(91,217,105,203),(92,223,106,209),(93,222,107,208),(94,221,108,207),(95,220,109,206),(96,219,110,205),(97,218,111,204),(98,224,112,210)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,119),(8,125),(9,124),(10,123),(11,122),(12,121),(13,120),(14,126),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,133),(22,139),(23,138),(24,137),(25,136),(26,135),(27,134),(28,140),(29,146),(30,145),(31,144),(32,143),(33,142),(34,141),(35,147),(36,153),(37,152),(38,151),(39,150),(40,149),(41,148),(42,154),(43,160),(44,159),(45,158),(46,157),(47,156),(48,155),(49,161),(50,167),(51,166),(52,165),(53,164),(54,163),(55,162),(56,168),(57,174),(58,173),(59,172),(60,171),(61,170),(62,169),(63,175),(64,181),(65,180),(66,179),(67,178),(68,177),(69,176),(70,182),(71,188),(72,187),(73,186),(74,185),(75,184),(76,183),(77,189),(78,195),(79,194),(80,193),(81,192),(82,191),(83,190),(84,196),(85,202),(86,201),(87,200),(88,199),(89,198),(90,197),(91,203),(92,209),(93,208),(94,207),(95,206),(96,205),(97,204),(98,210),(99,216),(100,215),(101,214),(102,213),(103,212),(104,211),(105,217),(106,223),(107,222),(108,221),(109,220),(110,219),(111,218),(112,224)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 2 | 2 | 4 | ··· | 4 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | C7⋊D4 | C7⋊D4 | C7⋊D4 | C8.C22 | Q8⋊D7 | C28.C23 |
| kernel | (C7×Q8)⋊13D4 | C28.55D4 | Q8⋊Dic7 | C28⋊7D4 | C2×Q8⋊D7 | Q8×C2×C14 | C2×C28 | C7×Q8 | C22×C14 | C22×Q8 | C2×C14 | C22×C4 | C2×Q8 | C2×C4 | Q8 | C23 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 3 | 4 | 3 | 6 | 6 | 24 | 6 | 1 | 6 | 6 |
Matrix representation of (C7×Q8)⋊13D4 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 78 |
| 0 | 0 | 84 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 65 |
| 0 | 0 | 94 | 68 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 16 | 96 |
| 0 | 0 | 15 | 97 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 96 |
| 0 | 0 | 15 | 97 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,109,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,84,0,0,78,112],[112,0,0,0,0,1,0,0,0,0,45,94,0,0,65,68],[0,112,0,0,1,0,0,0,0,0,16,15,0,0,96,97],[0,1,0,0,1,0,0,0,0,0,16,15,0,0,96,97] >;
(C7×Q8)⋊13D4 in GAP, Magma, Sage, TeX
(C_7\times Q_8)\rtimes_{13}D_4 % in TeX
G:=Group("(C7xQ8):13D4"); // GroupNames label
G:=SmallGroup(448,761);
// by ID
G=gap.SmallGroup(448,761);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations