metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C8⋊40C2, C4.89(C2×D28), (C2×D28).14C4, (C2×C8).189D14, (C2×C4).152D28, C28.444(C2×D4), (C2×C28).171D4, (C2×M4(2))⋊9D7, C23.29(C4×D7), C4.12(D14⋊C4), C14.34(C8○D4), C28.26(C22⋊C4), (C14×M4(2))⋊17C2, (C2×C28).869C23, C2.19(D28.C4), (C2×C56).319C22, C22.2(D14⋊C4), (C2×Dic14).14C4, (C22×C4).350D14, (C22×C28).186C22, (C22×C7⋊C8)⋊6C2, (C2×C4).84(C4×D7), C2.28(C2×D14⋊C4), (C2×C7⋊D4).12C4, C4.135(C2×C7⋊D4), C22.148(C2×C4×D7), (C2×C28).106(C2×C4), C7⋊3((C22×C8)⋊C2), (C2×C4○D28).11C2, C14.56(C2×C22⋊C4), (C2×C7⋊C8).325C22, (C2×C4×D7).186C22, (C2×C4).141(C7⋊D4), (C22×C14).69(C2×C4), (C2×Dic7).35(C2×C4), (C22×D7).26(C2×C4), (C2×C4).811(C22×D7), (C2×C14).20(C22⋊C4), (C2×C14).139(C22×C4), SmallGroup(448,663)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×M4(2))⋊D7
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=ab5, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 740 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D7 [×2], C14, C14 [×2], C14 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic7 [×2], C28 [×2], C28 [×2], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×2], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C7⋊C8 [×2], C56 [×2], Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7 [×2], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C22×D7 [×2], C22×C14, (C22×C8)⋊C2, C2×C7⋊C8 [×2], C2×C7⋊C8 [×2], C2×C56 [×2], C7×M4(2) [×2], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×4], C2×C7⋊D4 [×2], C22×C28, D14⋊C8 [×4], C22×C7⋊C8, C14×M4(2), C2×C4○D28, (C2×M4(2))⋊D7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D14 [×3], C2×C22⋊C4, C8○D4 [×2], C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C22×D7, (C22×C8)⋊C2, D14⋊C4 [×4], C2×C4×D7, C2×D28, C2×C7⋊D4, D28.C4 [×2], C2×D14⋊C4, (C2×M4(2))⋊D7
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 165)(10 166)(11 167)(12 168)(13 161)(14 162)(15 163)(16 164)(17 129)(18 130)(19 131)(20 132)(21 133)(22 134)(23 135)(24 136)(25 118)(26 119)(27 120)(28 113)(29 114)(30 115)(31 116)(32 117)(33 126)(34 127)(35 128)(36 121)(37 122)(38 123)(39 124)(40 125)(41 183)(42 184)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 151)(58 152)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 169)(72 170)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)(81 188)(82 189)(83 190)(84 191)(85 192)(86 185)(87 186)(88 187)(89 196)(90 197)(91 198)(92 199)(93 200)(94 193)(95 194)(96 195)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 201)(209 222)(210 223)(211 224)(212 217)(213 218)(214 219)(215 220)(216 221)
(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)
(1 103)(2 100)(3 97)(4 102)(5 99)(6 104)(7 101)(8 98)(9 161)(10 166)(11 163)(12 168)(13 165)(14 162)(15 167)(16 164)(17 129)(18 134)(19 131)(20 136)(21 133)(22 130)(23 135)(24 132)(25 114)(26 119)(27 116)(28 113)(29 118)(30 115)(31 120)(32 117)(33 126)(34 123)(35 128)(36 125)(37 122)(38 127)(39 124)(40 121)(41 179)(42 184)(43 181)(44 178)(45 183)(46 180)(47 177)(48 182)(49 139)(50 144)(51 141)(52 138)(53 143)(54 140)(55 137)(56 142)(57 151)(58 148)(59 145)(60 150)(61 147)(62 152)(63 149)(64 146)(65 171)(66 176)(67 173)(68 170)(69 175)(70 172)(71 169)(72 174)(73 108)(74 105)(75 110)(76 107)(77 112)(78 109)(79 106)(80 111)(81 192)(82 189)(83 186)(84 191)(85 188)(86 185)(87 190)(88 187)(89 196)(90 193)(91 198)(92 195)(93 200)(94 197)(95 194)(96 199)(153 202)(154 207)(155 204)(156 201)(157 206)(158 203)(159 208)(160 205)(209 218)(210 223)(211 220)(212 217)(213 222)(214 219)(215 224)(216 221)
(1 13 45 137 176 20 74)(2 14 46 138 169 21 75)(3 15 47 139 170 22 76)(4 16 48 140 171 23 77)(5 9 41 141 172 24 78)(6 10 42 142 173 17 79)(7 11 43 143 174 18 80)(8 12 44 144 175 19 73)(25 190 152 38 90 203 224)(26 191 145 39 91 204 217)(27 192 146 40 92 205 218)(28 185 147 33 93 206 219)(29 186 148 34 94 207 220)(30 187 149 35 95 208 221)(31 188 150 36 96 201 222)(32 189 151 37 89 202 223)(49 68 130 107 97 167 177)(50 69 131 108 98 168 178)(51 70 132 109 99 161 179)(52 71 133 110 100 162 180)(53 72 134 111 101 163 181)(54 65 135 112 102 164 182)(55 66 136 105 103 165 183)(56 67 129 106 104 166 184)(57 122 196 153 210 117 82)(58 123 197 154 211 118 83)(59 124 198 155 212 119 84)(60 125 199 156 213 120 85)(61 126 200 157 214 113 86)(62 127 193 158 215 114 87)(63 128 194 159 216 115 88)(64 121 195 160 209 116 81)
(1 91)(2 195)(3 93)(4 197)(5 95)(6 199)(7 89)(8 193)(9 35)(10 125)(11 37)(12 127)(13 39)(14 121)(15 33)(16 123)(17 213)(18 223)(19 215)(20 217)(21 209)(22 219)(23 211)(24 221)(25 65)(26 176)(27 67)(28 170)(29 69)(30 172)(31 71)(32 174)(34 168)(36 162)(38 164)(40 166)(41 149)(42 60)(43 151)(44 62)(45 145)(46 64)(47 147)(48 58)(49 82)(50 186)(51 84)(52 188)(53 86)(54 190)(55 88)(56 192)(57 177)(59 179)(61 181)(63 183)(66 115)(68 117)(70 119)(72 113)(73 158)(74 204)(75 160)(76 206)(77 154)(78 208)(79 156)(80 202)(81 138)(83 140)(85 142)(87 144)(90 102)(92 104)(94 98)(96 100)(97 196)(99 198)(101 200)(103 194)(105 159)(106 205)(107 153)(108 207)(109 155)(110 201)(111 157)(112 203)(114 175)(116 169)(118 171)(120 173)(122 167)(124 161)(126 163)(128 165)(129 218)(130 210)(131 220)(132 212)(133 222)(134 214)(135 224)(136 216)(137 191)(139 185)(141 187)(143 189)(146 184)(148 178)(150 180)(152 182)
G:=sub<Sym(224)| (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,165)(10,166)(11,167)(12,168)(13,161)(14,162)(15,163)(16,164)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,118)(26,119)(27,120)(28,113)(29,114)(30,115)(31,116)(32,117)(33,126)(34,127)(35,128)(36,121)(37,122)(38,123)(39,124)(40,125)(41,183)(42,184)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,143)(50,144)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,151)(58,152)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,169)(72,170)(73,108)(74,109)(75,110)(76,111)(77,112)(78,105)(79,106)(80,107)(81,188)(82,189)(83,190)(84,191)(85,192)(86,185)(87,186)(88,187)(89,196)(90,197)(91,198)(92,199)(93,200)(94,193)(95,194)(96,195)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,201)(209,222)(210,223)(211,224)(212,217)(213,218)(214,219)(215,220)(216,221), (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), (1,103)(2,100)(3,97)(4,102)(5,99)(6,104)(7,101)(8,98)(9,161)(10,166)(11,163)(12,168)(13,165)(14,162)(15,167)(16,164)(17,129)(18,134)(19,131)(20,136)(21,133)(22,130)(23,135)(24,132)(25,114)(26,119)(27,116)(28,113)(29,118)(30,115)(31,120)(32,117)(33,126)(34,123)(35,128)(36,125)(37,122)(38,127)(39,124)(40,121)(41,179)(42,184)(43,181)(44,178)(45,183)(46,180)(47,177)(48,182)(49,139)(50,144)(51,141)(52,138)(53,143)(54,140)(55,137)(56,142)(57,151)(58,148)(59,145)(60,150)(61,147)(62,152)(63,149)(64,146)(65,171)(66,176)(67,173)(68,170)(69,175)(70,172)(71,169)(72,174)(73,108)(74,105)(75,110)(76,107)(77,112)(78,109)(79,106)(80,111)(81,192)(82,189)(83,186)(84,191)(85,188)(86,185)(87,190)(88,187)(89,196)(90,193)(91,198)(92,195)(93,200)(94,197)(95,194)(96,199)(153,202)(154,207)(155,204)(156,201)(157,206)(158,203)(159,208)(160,205)(209,218)(210,223)(211,220)(212,217)(213,222)(214,219)(215,224)(216,221), (1,13,45,137,176,20,74)(2,14,46,138,169,21,75)(3,15,47,139,170,22,76)(4,16,48,140,171,23,77)(5,9,41,141,172,24,78)(6,10,42,142,173,17,79)(7,11,43,143,174,18,80)(8,12,44,144,175,19,73)(25,190,152,38,90,203,224)(26,191,145,39,91,204,217)(27,192,146,40,92,205,218)(28,185,147,33,93,206,219)(29,186,148,34,94,207,220)(30,187,149,35,95,208,221)(31,188,150,36,96,201,222)(32,189,151,37,89,202,223)(49,68,130,107,97,167,177)(50,69,131,108,98,168,178)(51,70,132,109,99,161,179)(52,71,133,110,100,162,180)(53,72,134,111,101,163,181)(54,65,135,112,102,164,182)(55,66,136,105,103,165,183)(56,67,129,106,104,166,184)(57,122,196,153,210,117,82)(58,123,197,154,211,118,83)(59,124,198,155,212,119,84)(60,125,199,156,213,120,85)(61,126,200,157,214,113,86)(62,127,193,158,215,114,87)(63,128,194,159,216,115,88)(64,121,195,160,209,116,81), (1,91)(2,195)(3,93)(4,197)(5,95)(6,199)(7,89)(8,193)(9,35)(10,125)(11,37)(12,127)(13,39)(14,121)(15,33)(16,123)(17,213)(18,223)(19,215)(20,217)(21,209)(22,219)(23,211)(24,221)(25,65)(26,176)(27,67)(28,170)(29,69)(30,172)(31,71)(32,174)(34,168)(36,162)(38,164)(40,166)(41,149)(42,60)(43,151)(44,62)(45,145)(46,64)(47,147)(48,58)(49,82)(50,186)(51,84)(52,188)(53,86)(54,190)(55,88)(56,192)(57,177)(59,179)(61,181)(63,183)(66,115)(68,117)(70,119)(72,113)(73,158)(74,204)(75,160)(76,206)(77,154)(78,208)(79,156)(80,202)(81,138)(83,140)(85,142)(87,144)(90,102)(92,104)(94,98)(96,100)(97,196)(99,198)(101,200)(103,194)(105,159)(106,205)(107,153)(108,207)(109,155)(110,201)(111,157)(112,203)(114,175)(116,169)(118,171)(120,173)(122,167)(124,161)(126,163)(128,165)(129,218)(130,210)(131,220)(132,212)(133,222)(134,214)(135,224)(136,216)(137,191)(139,185)(141,187)(143,189)(146,184)(148,178)(150,180)(152,182)>;
G:=Group( (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,165)(10,166)(11,167)(12,168)(13,161)(14,162)(15,163)(16,164)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,118)(26,119)(27,120)(28,113)(29,114)(30,115)(31,116)(32,117)(33,126)(34,127)(35,128)(36,121)(37,122)(38,123)(39,124)(40,125)(41,183)(42,184)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,143)(50,144)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,151)(58,152)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,169)(72,170)(73,108)(74,109)(75,110)(76,111)(77,112)(78,105)(79,106)(80,107)(81,188)(82,189)(83,190)(84,191)(85,192)(86,185)(87,186)(88,187)(89,196)(90,197)(91,198)(92,199)(93,200)(94,193)(95,194)(96,195)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,201)(209,222)(210,223)(211,224)(212,217)(213,218)(214,219)(215,220)(216,221), (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), (1,103)(2,100)(3,97)(4,102)(5,99)(6,104)(7,101)(8,98)(9,161)(10,166)(11,163)(12,168)(13,165)(14,162)(15,167)(16,164)(17,129)(18,134)(19,131)(20,136)(21,133)(22,130)(23,135)(24,132)(25,114)(26,119)(27,116)(28,113)(29,118)(30,115)(31,120)(32,117)(33,126)(34,123)(35,128)(36,125)(37,122)(38,127)(39,124)(40,121)(41,179)(42,184)(43,181)(44,178)(45,183)(46,180)(47,177)(48,182)(49,139)(50,144)(51,141)(52,138)(53,143)(54,140)(55,137)(56,142)(57,151)(58,148)(59,145)(60,150)(61,147)(62,152)(63,149)(64,146)(65,171)(66,176)(67,173)(68,170)(69,175)(70,172)(71,169)(72,174)(73,108)(74,105)(75,110)(76,107)(77,112)(78,109)(79,106)(80,111)(81,192)(82,189)(83,186)(84,191)(85,188)(86,185)(87,190)(88,187)(89,196)(90,193)(91,198)(92,195)(93,200)(94,197)(95,194)(96,199)(153,202)(154,207)(155,204)(156,201)(157,206)(158,203)(159,208)(160,205)(209,218)(210,223)(211,220)(212,217)(213,222)(214,219)(215,224)(216,221), (1,13,45,137,176,20,74)(2,14,46,138,169,21,75)(3,15,47,139,170,22,76)(4,16,48,140,171,23,77)(5,9,41,141,172,24,78)(6,10,42,142,173,17,79)(7,11,43,143,174,18,80)(8,12,44,144,175,19,73)(25,190,152,38,90,203,224)(26,191,145,39,91,204,217)(27,192,146,40,92,205,218)(28,185,147,33,93,206,219)(29,186,148,34,94,207,220)(30,187,149,35,95,208,221)(31,188,150,36,96,201,222)(32,189,151,37,89,202,223)(49,68,130,107,97,167,177)(50,69,131,108,98,168,178)(51,70,132,109,99,161,179)(52,71,133,110,100,162,180)(53,72,134,111,101,163,181)(54,65,135,112,102,164,182)(55,66,136,105,103,165,183)(56,67,129,106,104,166,184)(57,122,196,153,210,117,82)(58,123,197,154,211,118,83)(59,124,198,155,212,119,84)(60,125,199,156,213,120,85)(61,126,200,157,214,113,86)(62,127,193,158,215,114,87)(63,128,194,159,216,115,88)(64,121,195,160,209,116,81), (1,91)(2,195)(3,93)(4,197)(5,95)(6,199)(7,89)(8,193)(9,35)(10,125)(11,37)(12,127)(13,39)(14,121)(15,33)(16,123)(17,213)(18,223)(19,215)(20,217)(21,209)(22,219)(23,211)(24,221)(25,65)(26,176)(27,67)(28,170)(29,69)(30,172)(31,71)(32,174)(34,168)(36,162)(38,164)(40,166)(41,149)(42,60)(43,151)(44,62)(45,145)(46,64)(47,147)(48,58)(49,82)(50,186)(51,84)(52,188)(53,86)(54,190)(55,88)(56,192)(57,177)(59,179)(61,181)(63,183)(66,115)(68,117)(70,119)(72,113)(73,158)(74,204)(75,160)(76,206)(77,154)(78,208)(79,156)(80,202)(81,138)(83,140)(85,142)(87,144)(90,102)(92,104)(94,98)(96,100)(97,196)(99,198)(101,200)(103,194)(105,159)(106,205)(107,153)(108,207)(109,155)(110,201)(111,157)(112,203)(114,175)(116,169)(118,171)(120,173)(122,167)(124,161)(126,163)(128,165)(129,218)(130,210)(131,220)(132,212)(133,222)(134,214)(135,224)(136,216)(137,191)(139,185)(141,187)(143,189)(146,184)(148,178)(150,180)(152,182) );
G=PermutationGroup([(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,97),(8,98),(9,165),(10,166),(11,167),(12,168),(13,161),(14,162),(15,163),(16,164),(17,129),(18,130),(19,131),(20,132),(21,133),(22,134),(23,135),(24,136),(25,118),(26,119),(27,120),(28,113),(29,114),(30,115),(31,116),(32,117),(33,126),(34,127),(35,128),(36,121),(37,122),(38,123),(39,124),(40,125),(41,183),(42,184),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,143),(50,144),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,151),(58,152),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,171),(66,172),(67,173),(68,174),(69,175),(70,176),(71,169),(72,170),(73,108),(74,109),(75,110),(76,111),(77,112),(78,105),(79,106),(80,107),(81,188),(82,189),(83,190),(84,191),(85,192),(86,185),(87,186),(88,187),(89,196),(90,197),(91,198),(92,199),(93,200),(94,193),(95,194),(96,195),(153,202),(154,203),(155,204),(156,205),(157,206),(158,207),(159,208),(160,201),(209,222),(210,223),(211,224),(212,217),(213,218),(214,219),(215,220),(216,221)], [(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)], [(1,103),(2,100),(3,97),(4,102),(5,99),(6,104),(7,101),(8,98),(9,161),(10,166),(11,163),(12,168),(13,165),(14,162),(15,167),(16,164),(17,129),(18,134),(19,131),(20,136),(21,133),(22,130),(23,135),(24,132),(25,114),(26,119),(27,116),(28,113),(29,118),(30,115),(31,120),(32,117),(33,126),(34,123),(35,128),(36,125),(37,122),(38,127),(39,124),(40,121),(41,179),(42,184),(43,181),(44,178),(45,183),(46,180),(47,177),(48,182),(49,139),(50,144),(51,141),(52,138),(53,143),(54,140),(55,137),(56,142),(57,151),(58,148),(59,145),(60,150),(61,147),(62,152),(63,149),(64,146),(65,171),(66,176),(67,173),(68,170),(69,175),(70,172),(71,169),(72,174),(73,108),(74,105),(75,110),(76,107),(77,112),(78,109),(79,106),(80,111),(81,192),(82,189),(83,186),(84,191),(85,188),(86,185),(87,190),(88,187),(89,196),(90,193),(91,198),(92,195),(93,200),(94,197),(95,194),(96,199),(153,202),(154,207),(155,204),(156,201),(157,206),(158,203),(159,208),(160,205),(209,218),(210,223),(211,220),(212,217),(213,222),(214,219),(215,224),(216,221)], [(1,13,45,137,176,20,74),(2,14,46,138,169,21,75),(3,15,47,139,170,22,76),(4,16,48,140,171,23,77),(5,9,41,141,172,24,78),(6,10,42,142,173,17,79),(7,11,43,143,174,18,80),(8,12,44,144,175,19,73),(25,190,152,38,90,203,224),(26,191,145,39,91,204,217),(27,192,146,40,92,205,218),(28,185,147,33,93,206,219),(29,186,148,34,94,207,220),(30,187,149,35,95,208,221),(31,188,150,36,96,201,222),(32,189,151,37,89,202,223),(49,68,130,107,97,167,177),(50,69,131,108,98,168,178),(51,70,132,109,99,161,179),(52,71,133,110,100,162,180),(53,72,134,111,101,163,181),(54,65,135,112,102,164,182),(55,66,136,105,103,165,183),(56,67,129,106,104,166,184),(57,122,196,153,210,117,82),(58,123,197,154,211,118,83),(59,124,198,155,212,119,84),(60,125,199,156,213,120,85),(61,126,200,157,214,113,86),(62,127,193,158,215,114,87),(63,128,194,159,216,115,88),(64,121,195,160,209,116,81)], [(1,91),(2,195),(3,93),(4,197),(5,95),(6,199),(7,89),(8,193),(9,35),(10,125),(11,37),(12,127),(13,39),(14,121),(15,33),(16,123),(17,213),(18,223),(19,215),(20,217),(21,209),(22,219),(23,211),(24,221),(25,65),(26,176),(27,67),(28,170),(29,69),(30,172),(31,71),(32,174),(34,168),(36,162),(38,164),(40,166),(41,149),(42,60),(43,151),(44,62),(45,145),(46,64),(47,147),(48,58),(49,82),(50,186),(51,84),(52,188),(53,86),(54,190),(55,88),(56,192),(57,177),(59,179),(61,181),(63,183),(66,115),(68,117),(70,119),(72,113),(73,158),(74,204),(75,160),(76,206),(77,154),(78,208),(79,156),(80,202),(81,138),(83,140),(85,142),(87,144),(90,102),(92,104),(94,98),(96,100),(97,196),(99,198),(101,200),(103,194),(105,159),(106,205),(107,153),(108,207),(109,155),(110,201),(111,157),(112,203),(114,175),(116,169),(118,171),(120,173),(122,167),(124,161),(126,163),(128,165),(129,218),(130,210),(131,220),(132,212),(133,222),(134,214),(135,224),(136,216),(137,191),(139,185),(141,187),(143,189),(146,184),(148,178),(150,180),(152,182)])
88 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | ||||||||
image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | D14 | D14 | C8○D4 | C4×D7 | D28 | C7⋊D4 | C4×D7 | D28.C4 |
kernel | (C2×M4(2))⋊D7 | D14⋊C8 | C22×C7⋊C8 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C2×C7⋊D4 | C2×C28 | C2×M4(2) | C2×C8 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
# reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 3 | 6 | 3 | 8 | 6 | 12 | 12 | 6 | 12 |
Matrix representation of (C2×M4(2))⋊D7 ►in GL4(𝔽113) generated by
112 | 0 | 0 | 0 |
0 | 112 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
58 | 75 | 0 | 0 |
38 | 55 | 0 | 0 |
0 | 0 | 98 | 2 |
0 | 0 | 106 | 15 |
112 | 0 | 0 | 0 |
0 | 112 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 15 | 112 |
0 | 1 | 0 | 0 |
112 | 9 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
104 | 1 | 0 | 0 |
33 | 9 | 0 | 0 |
0 | 0 | 69 | 36 |
0 | 0 | 31 | 44 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[58,38,0,0,75,55,0,0,0,0,98,106,0,0,2,15],[112,0,0,0,0,112,0,0,0,0,1,15,0,0,0,112],[0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[104,33,0,0,1,9,0,0,0,0,69,31,0,0,36,44] >;
(C2×M4(2))⋊D7 in GAP, Magma, Sage, TeX
(C_2\times M_{4(2}))\rtimes D_7
% in TeX
G:=Group("(C2xM4(2)):D7");
// GroupNames label
G:=SmallGroup(448,663);
// by ID
G=gap.SmallGroup(448,663);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,422,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=a*b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations