metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.12D14, (C2×C28)⋊20D4, C23⋊2(C4×D7), C14.64(C4×D4), (C2×Dic7)⋊16D4, C14.38C22≀C2, D14⋊2(C22⋊C4), C2.2(C28⋊2D4), (C22×D7).88D4, (C22×C4).31D14, C22.101(D4×D7), C2.3(C23⋊D14), C2.6(D14⋊D4), C14.31(C4⋊D4), C7⋊2(C23.23D4), C14.C42⋊39C2, C2.7(D14.D4), C22.53(C4○D28), (C23×C14).39C22, (C23×D7).88C22, C23.283(C22×D7), C2.27(Dic7⋊4D4), C22.48(D4⋊2D7), (C22×C28).344C22, (C22×C14).330C23, C14.32(C22.D4), (C22×Dic7).43C22, (C2×C7⋊D4)⋊4C4, C2.9(C4×C7⋊D4), (C2×D14⋊C4)⋊4C2, (C2×C22⋊C4)⋊3D7, (D7×C22×C4)⋊13C2, (C2×C4)⋊12(C7⋊D4), (C22×C14)⋊6(C2×C4), (C2×Dic7)⋊5(C2×C4), (C2×C23.D7)⋊3C2, C2.29(D7×C22⋊C4), (C14×C22⋊C4)⋊22C2, C22.127(C2×C4×D7), (C2×C14).322(C2×D4), C14.28(C2×C22⋊C4), (C22×C7⋊D4).2C2, C22.51(C2×C7⋊D4), (C22×D7).41(C2×C4), (C2×C14).145(C4○D4), (C2×C14).109(C22×C4), SmallGroup(448,490)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.12D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1428 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23.23D4, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C14.C42, C2×D14⋊C4, C2×C23.D7, C14×C22⋊C4, D7×C22×C4, C22×C7⋊D4, C24.12D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C4○D4, D14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D7, C7⋊D4, C22×D7, C23.23D4, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, D7×C22⋊C4, Dic7⋊4D4, D14.D4, D14⋊D4, C4×C7⋊D4, C23⋊D14, C28⋊2D4, C24.12D14
►On 224 points
Generators in S224
(1 194)(2 62)(3 196)(4 64)(5 170)(6 66)(7 172)(8 68)(9 174)(10 70)(11 176)(12 72)(13 178)(14 74)(15 180)(16 76)(17 182)(18 78)(19 184)(20 80)(21 186)(22 82)(23 188)(24 84)(25 190)(26 58)(27 192)(28 60)(29 105)(30 198)(31 107)(32 200)(33 109)(34 202)(35 111)(36 204)(37 85)(38 206)(39 87)(40 208)(41 89)(42 210)(43 91)(44 212)(45 93)(46 214)(47 95)(48 216)(49 97)(50 218)(51 99)(52 220)(53 101)(54 222)(55 103)(56 224)(57 163)(59 165)(61 167)(63 141)(65 143)(67 145)(69 147)(71 149)(73 151)(75 153)(77 155)(79 157)(81 159)(83 161)(86 121)(88 123)(90 125)(92 127)(94 129)(96 131)(98 133)(100 135)(102 137)(104 139)(106 113)(108 115)(110 117)(112 119)(114 199)(116 201)(118 203)(120 205)(122 207)(124 209)(126 211)(128 213)(130 215)(132 217)(134 219)(136 221)(138 223)(140 197)(142 169)(144 171)(146 173)(148 175)(150 177)(152 179)(154 181)(156 183)(158 185)(160 187)(162 189)(164 191)(166 193)(168 195)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 222)(2 223)(3 224)(4 197)(5 198)(6 199)(7 200)(8 201)(9 202)(10 203)(11 204)(12 205)(13 206)(14 207)(15 208)(16 209)(17 210)(18 211)(19 212)(20 213)(21 214)(22 215)(23 216)(24 217)(25 218)(26 219)(27 220)(28 221)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 161)(97 162)(98 163)(99 164)(100 165)(101 166)(102 167)(103 168)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)
(1 167)(2 168)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 159)(22 160)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 140)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 197)(106 198)(107 199)(108 200)(109 201)(110 202)(111 203)(112 204)
(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)
(1 166 15 152)(2 151 16 165)(3 164 17 150)(4 149 18 163)(5 162 19 148)(6 147 20 161)(7 160 21 146)(8 145 22 159)(9 158 23 144)(10 143 24 157)(11 156 25 142)(12 141 26 155)(13 154 27 168)(14 167 28 153)(29 71 43 57)(30 84 44 70)(31 69 45 83)(32 82 46 68)(33 67 47 81)(34 80 48 66)(35 65 49 79)(36 78 50 64)(37 63 51 77)(38 76 52 62)(39 61 53 75)(40 74 54 60)(41 59 55 73)(42 72 56 58)(85 224 99 210)(86 209 100 223)(87 222 101 208)(88 207 102 221)(89 220 103 206)(90 205 104 219)(91 218 105 204)(92 203 106 217)(93 216 107 202)(94 201 108 215)(95 214 109 200)(96 199 110 213)(97 212 111 198)(98 197 112 211)(113 189 127 175)(114 174 128 188)(115 187 129 173)(116 172 130 186)(117 185 131 171)(118 170 132 184)(119 183 133 169)(120 196 134 182)(121 181 135 195)(122 194 136 180)(123 179 137 193)(124 192 138 178)(125 177 139 191)(126 190 140 176)
G:=sub<Sym(224)| (1,194)(2,62)(3,196)(4,64)(5,170)(6,66)(7,172)(8,68)(9,174)(10,70)(11,176)(12,72)(13,178)(14,74)(15,180)(16,76)(17,182)(18,78)(19,184)(20,80)(21,186)(22,82)(23,188)(24,84)(25,190)(26,58)(27,192)(28,60)(29,105)(30,198)(31,107)(32,200)(33,109)(34,202)(35,111)(36,204)(37,85)(38,206)(39,87)(40,208)(41,89)(42,210)(43,91)(44,212)(45,93)(46,214)(47,95)(48,216)(49,97)(50,218)(51,99)(52,220)(53,101)(54,222)(55,103)(56,224)(57,163)(59,165)(61,167)(63,141)(65,143)(67,145)(69,147)(71,149)(73,151)(75,153)(77,155)(79,157)(81,159)(83,161)(86,121)(88,123)(90,125)(92,127)(94,129)(96,131)(98,133)(100,135)(102,137)(104,139)(106,113)(108,115)(110,117)(112,119)(114,199)(116,201)(118,203)(120,205)(122,207)(124,209)(126,211)(128,213)(130,215)(132,217)(134,219)(136,221)(138,223)(140,197)(142,169)(144,171)(146,173)(148,175)(150,177)(152,179)(154,181)(156,183)(158,185)(160,187)(162,189)(164,191)(166,193)(168,195), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,222)(2,223)(3,224)(4,197)(5,198)(6,199)(7,200)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149), (1,167)(2,168)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,140)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)(111,203)(112,204), (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), (1,166,15,152)(2,151,16,165)(3,164,17,150)(4,149,18,163)(5,162,19,148)(6,147,20,161)(7,160,21,146)(8,145,22,159)(9,158,23,144)(10,143,24,157)(11,156,25,142)(12,141,26,155)(13,154,27,168)(14,167,28,153)(29,71,43,57)(30,84,44,70)(31,69,45,83)(32,82,46,68)(33,67,47,81)(34,80,48,66)(35,65,49,79)(36,78,50,64)(37,63,51,77)(38,76,52,62)(39,61,53,75)(40,74,54,60)(41,59,55,73)(42,72,56,58)(85,224,99,210)(86,209,100,223)(87,222,101,208)(88,207,102,221)(89,220,103,206)(90,205,104,219)(91,218,105,204)(92,203,106,217)(93,216,107,202)(94,201,108,215)(95,214,109,200)(96,199,110,213)(97,212,111,198)(98,197,112,211)(113,189,127,175)(114,174,128,188)(115,187,129,173)(116,172,130,186)(117,185,131,171)(118,170,132,184)(119,183,133,169)(120,196,134,182)(121,181,135,195)(122,194,136,180)(123,179,137,193)(124,192,138,178)(125,177,139,191)(126,190,140,176)>;
G:=Group( (1,194)(2,62)(3,196)(4,64)(5,170)(6,66)(7,172)(8,68)(9,174)(10,70)(11,176)(12,72)(13,178)(14,74)(15,180)(16,76)(17,182)(18,78)(19,184)(20,80)(21,186)(22,82)(23,188)(24,84)(25,190)(26,58)(27,192)(28,60)(29,105)(30,198)(31,107)(32,200)(33,109)(34,202)(35,111)(36,204)(37,85)(38,206)(39,87)(40,208)(41,89)(42,210)(43,91)(44,212)(45,93)(46,214)(47,95)(48,216)(49,97)(50,218)(51,99)(52,220)(53,101)(54,222)(55,103)(56,224)(57,163)(59,165)(61,167)(63,141)(65,143)(67,145)(69,147)(71,149)(73,151)(75,153)(77,155)(79,157)(81,159)(83,161)(86,121)(88,123)(90,125)(92,127)(94,129)(96,131)(98,133)(100,135)(102,137)(104,139)(106,113)(108,115)(110,117)(112,119)(114,199)(116,201)(118,203)(120,205)(122,207)(124,209)(126,211)(128,213)(130,215)(132,217)(134,219)(136,221)(138,223)(140,197)(142,169)(144,171)(146,173)(148,175)(150,177)(152,179)(154,181)(156,183)(158,185)(160,187)(162,189)(164,191)(166,193)(168,195), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,222)(2,223)(3,224)(4,197)(5,198)(6,199)(7,200)(8,201)(9,202)(10,203)(11,204)(12,205)(13,206)(14,207)(15,208)(16,209)(17,210)(18,211)(19,212)(20,213)(21,214)(22,215)(23,216)(24,217)(25,218)(26,219)(27,220)(28,221)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149), (1,167)(2,168)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,140)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)(111,203)(112,204), (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), (1,166,15,152)(2,151,16,165)(3,164,17,150)(4,149,18,163)(5,162,19,148)(6,147,20,161)(7,160,21,146)(8,145,22,159)(9,158,23,144)(10,143,24,157)(11,156,25,142)(12,141,26,155)(13,154,27,168)(14,167,28,153)(29,71,43,57)(30,84,44,70)(31,69,45,83)(32,82,46,68)(33,67,47,81)(34,80,48,66)(35,65,49,79)(36,78,50,64)(37,63,51,77)(38,76,52,62)(39,61,53,75)(40,74,54,60)(41,59,55,73)(42,72,56,58)(85,224,99,210)(86,209,100,223)(87,222,101,208)(88,207,102,221)(89,220,103,206)(90,205,104,219)(91,218,105,204)(92,203,106,217)(93,216,107,202)(94,201,108,215)(95,214,109,200)(96,199,110,213)(97,212,111,198)(98,197,112,211)(113,189,127,175)(114,174,128,188)(115,187,129,173)(116,172,130,186)(117,185,131,171)(118,170,132,184)(119,183,133,169)(120,196,134,182)(121,181,135,195)(122,194,136,180)(123,179,137,193)(124,192,138,178)(125,177,139,191)(126,190,140,176) );
G=PermutationGroup([[(1,194),(2,62),(3,196),(4,64),(5,170),(6,66),(7,172),(8,68),(9,174),(10,70),(11,176),(12,72),(13,178),(14,74),(15,180),(16,76),(17,182),(18,78),(19,184),(20,80),(21,186),(22,82),(23,188),(24,84),(25,190),(26,58),(27,192),(28,60),(29,105),(30,198),(31,107),(32,200),(33,109),(34,202),(35,111),(36,204),(37,85),(38,206),(39,87),(40,208),(41,89),(42,210),(43,91),(44,212),(45,93),(46,214),(47,95),(48,216),(49,97),(50,218),(51,99),(52,220),(53,101),(54,222),(55,103),(56,224),(57,163),(59,165),(61,167),(63,141),(65,143),(67,145),(69,147),(71,149),(73,151),(75,153),(77,155),(79,157),(81,159),(83,161),(86,121),(88,123),(90,125),(92,127),(94,129),(96,131),(98,133),(100,135),(102,137),(104,139),(106,113),(108,115),(110,117),(112,119),(114,199),(116,201),(118,203),(120,205),(122,207),(124,209),(126,211),(128,213),(130,215),(132,217),(134,219),(136,221),(138,223),(140,197),(142,169),(144,171),(146,173),(148,175),(150,177),(152,179),(154,181),(156,183),(158,185),(160,187),(162,189),(164,191),(166,193),(168,195)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,222),(2,223),(3,224),(4,197),(5,198),(6,199),(7,200),(8,201),(9,202),(10,203),(11,204),(12,205),(13,206),(14,207),(15,208),(16,209),(17,210),(18,211),(19,212),(20,213),(21,214),(22,215),(23,216),(24,217),(25,218),(26,219),(27,220),(28,221),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,193),(54,194),(55,195),(56,196),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,140),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,161),(97,162),(98,163),(99,164),(100,165),(101,166),(102,167),(103,168),(104,141),(105,142),(106,143),(107,144),(108,145),(109,146),(110,147),(111,148),(112,149)], [(1,167),(2,168),(3,141),(4,142),(5,143),(6,144),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,153),(16,154),(17,155),(18,156),(19,157),(20,158),(21,159),(22,160),(23,161),(24,162),(25,163),(26,164),(27,165),(28,166),(29,140),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,120),(38,121),(39,122),(40,123),(41,124),(42,125),(43,126),(44,127),(45,128),(46,129),(47,130),(48,131),(49,132),(50,133),(51,134),(52,135),(53,136),(54,137),(55,138),(56,139),(57,190),(58,191),(59,192),(60,193),(61,194),(62,195),(63,196),(64,169),(65,170),(66,171),(67,172),(68,173),(69,174),(70,175),(71,176),(72,177),(73,178),(74,179),(75,180),(76,181),(77,182),(78,183),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,197),(106,198),(107,199),(108,200),(109,201),(110,202),(111,203),(112,204)], [(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)], [(1,166,15,152),(2,151,16,165),(3,164,17,150),(4,149,18,163),(5,162,19,148),(6,147,20,161),(7,160,21,146),(8,145,22,159),(9,158,23,144),(10,143,24,157),(11,156,25,142),(12,141,26,155),(13,154,27,168),(14,167,28,153),(29,71,43,57),(30,84,44,70),(31,69,45,83),(32,82,46,68),(33,67,47,81),(34,80,48,66),(35,65,49,79),(36,78,50,64),(37,63,51,77),(38,76,52,62),(39,61,53,75),(40,74,54,60),(41,59,55,73),(42,72,56,58),(85,224,99,210),(86,209,100,223),(87,222,101,208),(88,207,102,221),(89,220,103,206),(90,205,104,219),(91,218,105,204),(92,203,106,217),(93,216,107,202),(94,201,108,215),(95,214,109,200),(96,199,110,213),(97,212,111,198),(98,197,112,211),(113,189,127,175),(114,174,128,188),(115,187,129,173),(116,172,130,186),(117,185,131,171),(118,170,132,184),(119,183,133,169),(120,196,134,182),(121,181,135,195),(122,194,136,180),(123,179,137,193),(124,192,138,178),(125,177,139,191),(126,190,140,176)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 |
| kernel | C24.12D14 | C14.C42 | C2×D14⋊C4 | C2×C23.D7 | C14×C22⋊C4 | D7×C22×C4 | C22×C7⋊D4 | C2×C7⋊D4 | C2×Dic7 | C2×C28 | C22×D7 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 3 | 4 | 6 | 3 | 12 | 12 | 12 | 9 | 3 |
Matrix representation of C24.12D14 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 24 | 14 |
| 0 | 0 | 0 | 19 | 5 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 |
| 0 | 0 | 0 | 3 | 26 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 3 | 19 |
| 0 | 0 | 0 | 24 | 26 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,24,19,0,0,0,14,5],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,17,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,19,26],[12,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,3,24,0,0,0,19,26] >;
C24.12D14 in GAP, Magma, Sage, TeX
C_2^4._{12}D_{14} % in TeX
G:=Group("C2^4.12D14"); // GroupNames label
G:=SmallGroup(448,490);
// by ID
G=gap.SmallGroup(448,490);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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=e^13>;
// generators/relations