metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.4D14, C14.42(C4×D4), C23.D7⋊9C4, (C2×C28).248D4, C22.96(D4×D7), C23.16(C4×D7), (C22×C4).25D14, C14.83(C4⋊D4), (C2×Dic7).172D4, C14.C42⋊11C2, C2.2(Dic7⋊D4), C2.5(D14.D4), C14.37(C4.4D4), C2.1(C28.17D4), C22.51(C4○D28), (C23×C14).28C22, C7⋊5(C24.C22), C23.278(C22×D7), C14.27(C42⋊C2), C14.12(C42⋊2C2), C2.25(Dic7⋊4D4), C22.44(D4⋊2D7), (C22×C14).320C23, (C22×C28).342C22, C2.6(C23.D14), C14.28(C22.D4), (C22×Dic7).36C22, C2.14(C23.11D14), C2.8(C4×C7⋊D4), (C2×C4×Dic7)⋊22C2, (C2×C22⋊C4).7D7, C22.124(C2×C4×D7), (C2×Dic7⋊C4)⋊32C2, (C2×C14).430(C2×D4), (C2×C4).98(C7⋊D4), C22.48(C2×C7⋊D4), (C2×C23.D7).7C2, (C2×C14).75(C4○D4), (C14×C22⋊C4).24C2, (C22×C14).48(C2×C4), (C2×Dic7).59(C2×C4), (C2×C14).106(C22×C4), SmallGroup(448,479)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.4D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=db=bd, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 724 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C24.C22, C4×Dic7, Dic7⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C23.D7, C14×C22⋊C4, C24.4D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4×D7, C7⋊D4, C22×D7, C24.C22, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, C23.11D14, C23.D14, Dic7⋊4D4, D14.D4, C4×C7⋊D4, C28.17D4, Dic7⋊D4, C24.4D14
►On 224 points
Generators in S224
(2 134)(4 136)(6 138)(8 140)(10 114)(12 116)(14 118)(16 120)(18 122)(20 124)(22 126)(24 128)(26 130)(28 132)(30 165)(32 167)(34 141)(36 143)(38 145)(40 147)(42 149)(44 151)(46 153)(48 155)(50 157)(52 159)(54 161)(56 163)(57 180)(58 203)(59 182)(60 205)(61 184)(62 207)(63 186)(64 209)(65 188)(66 211)(67 190)(68 213)(69 192)(70 215)(71 194)(72 217)(73 196)(74 219)(75 170)(76 221)(77 172)(78 223)(79 174)(80 197)(81 176)(82 199)(83 178)(84 201)(85 222)(86 173)(87 224)(88 175)(89 198)(90 177)(91 200)(92 179)(93 202)(94 181)(95 204)(96 183)(97 206)(98 185)(99 208)(100 187)(101 210)(102 189)(103 212)(104 191)(105 214)(106 193)(107 216)(108 195)(109 218)(110 169)(111 220)(112 171)
(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)
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 164)(30 165)(31 166)(32 167)(33 168)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(41 148)(42 149)(43 150)(44 151)(45 152)(46 153)(47 154)(48 155)(49 156)(50 157)(51 158)(52 159)(53 160)(54 161)(55 162)(56 163)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)(169 219)(170 220)(171 221)(172 222)(173 223)(174 224)(175 197)(176 198)(177 199)(178 200)(179 201)(180 202)(181 203)(182 204)(183 205)(184 206)(185 207)(186 208)(187 209)(188 210)(189 211)(190 212)(191 213)(192 214)(193 215)(194 216)(195 217)(196 218)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 216)(58 217)(59 218)(60 219)(61 220)(62 221)(63 222)(64 223)(65 224)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(73 204)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 186)(86 187)(87 188)(88 189)(89 190)(90 191)(91 192)(92 193)(93 194)(94 195)(95 196)(96 169)(97 170)(98 171)(99 172)(100 173)(101 174)(102 175)(103 176)(104 177)(105 178)(106 179)(107 180)(108 181)(109 182)(110 183)(111 184)(112 185)(113 144)(114 145)(115 146)(116 147)(117 148)(118 149)(119 150)(120 151)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 141)(139 142)(140 143)
(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)
(1 224 43 79)(2 209 44 64)(3 222 45 77)(4 207 46 62)(5 220 47 75)(6 205 48 60)(7 218 49 73)(8 203 50 58)(9 216 51 71)(10 201 52 84)(11 214 53 69)(12 199 54 82)(13 212 55 67)(14 197 56 80)(15 210 29 65)(16 223 30 78)(17 208 31 63)(18 221 32 76)(19 206 33 61)(20 219 34 74)(21 204 35 59)(22 217 36 72)(23 202 37 57)(24 215 38 70)(25 200 39 83)(26 213 40 68)(27 198 41 81)(28 211 42 66)(85 135 172 152)(86 120 173 165)(87 133 174 150)(88 118 175 163)(89 131 176 148)(90 116 177 161)(91 129 178 146)(92 114 179 159)(93 127 180 144)(94 140 181 157)(95 125 182 142)(96 138 183 155)(97 123 184 168)(98 136 185 153)(99 121 186 166)(100 134 187 151)(101 119 188 164)(102 132 189 149)(103 117 190 162)(104 130 191 147)(105 115 192 160)(106 128 193 145)(107 113 194 158)(108 126 195 143)(109 139 196 156)(110 124 169 141)(111 137 170 154)(112 122 171 167)
G:=sub<Sym(224)| (2,134)(4,136)(6,138)(8,140)(10,114)(12,116)(14,118)(16,120)(18,122)(20,124)(22,126)(24,128)(26,130)(28,132)(30,165)(32,167)(34,141)(36,143)(38,145)(40,147)(42,149)(44,151)(46,153)(48,155)(50,157)(52,159)(54,161)(56,163)(57,180)(58,203)(59,182)(60,205)(61,184)(62,207)(63,186)(64,209)(65,188)(66,211)(67,190)(68,213)(69,192)(70,215)(71,194)(72,217)(73,196)(74,219)(75,170)(76,221)(77,172)(78,223)(79,174)(80,197)(81,176)(82,199)(83,178)(84,201)(85,222)(86,173)(87,224)(88,175)(89,198)(90,177)(91,200)(92,179)(93,202)(94,181)(95,204)(96,183)(97,206)(98,185)(99,208)(100,187)(101,210)(102,189)(103,212)(104,191)(105,214)(106,193)(107,216)(108,195)(109,218)(110,169)(111,220)(112,171), (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), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,164)(30,165)(31,166)(32,167)(33,168)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92)(169,219)(170,220)(171,221)(172,222)(173,223)(174,224)(175,197)(176,198)(177,199)(178,200)(179,201)(180,202)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)(189,211)(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)(196,218), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,216)(58,217)(59,218)(60,219)(61,220)(62,221)(63,222)(64,223)(65,224)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,186)(86,187)(87,188)(88,189)(89,190)(90,191)(91,192)(92,193)(93,194)(94,195)(95,196)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,141)(139,142)(140,143), (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), (1,224,43,79)(2,209,44,64)(3,222,45,77)(4,207,46,62)(5,220,47,75)(6,205,48,60)(7,218,49,73)(8,203,50,58)(9,216,51,71)(10,201,52,84)(11,214,53,69)(12,199,54,82)(13,212,55,67)(14,197,56,80)(15,210,29,65)(16,223,30,78)(17,208,31,63)(18,221,32,76)(19,206,33,61)(20,219,34,74)(21,204,35,59)(22,217,36,72)(23,202,37,57)(24,215,38,70)(25,200,39,83)(26,213,40,68)(27,198,41,81)(28,211,42,66)(85,135,172,152)(86,120,173,165)(87,133,174,150)(88,118,175,163)(89,131,176,148)(90,116,177,161)(91,129,178,146)(92,114,179,159)(93,127,180,144)(94,140,181,157)(95,125,182,142)(96,138,183,155)(97,123,184,168)(98,136,185,153)(99,121,186,166)(100,134,187,151)(101,119,188,164)(102,132,189,149)(103,117,190,162)(104,130,191,147)(105,115,192,160)(106,128,193,145)(107,113,194,158)(108,126,195,143)(109,139,196,156)(110,124,169,141)(111,137,170,154)(112,122,171,167)>;
G:=Group( (2,134)(4,136)(6,138)(8,140)(10,114)(12,116)(14,118)(16,120)(18,122)(20,124)(22,126)(24,128)(26,130)(28,132)(30,165)(32,167)(34,141)(36,143)(38,145)(40,147)(42,149)(44,151)(46,153)(48,155)(50,157)(52,159)(54,161)(56,163)(57,180)(58,203)(59,182)(60,205)(61,184)(62,207)(63,186)(64,209)(65,188)(66,211)(67,190)(68,213)(69,192)(70,215)(71,194)(72,217)(73,196)(74,219)(75,170)(76,221)(77,172)(78,223)(79,174)(80,197)(81,176)(82,199)(83,178)(84,201)(85,222)(86,173)(87,224)(88,175)(89,198)(90,177)(91,200)(92,179)(93,202)(94,181)(95,204)(96,183)(97,206)(98,185)(99,208)(100,187)(101,210)(102,189)(103,212)(104,191)(105,214)(106,193)(107,216)(108,195)(109,218)(110,169)(111,220)(112,171), (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), (1,133)(2,134)(3,135)(4,136)(5,137)(6,138)(7,139)(8,140)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,164)(30,165)(31,166)(32,167)(33,168)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,148)(42,149)(43,150)(44,151)(45,152)(46,153)(47,154)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92)(169,219)(170,220)(171,221)(172,222)(173,223)(174,224)(175,197)(176,198)(177,199)(178,200)(179,201)(180,202)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)(189,211)(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)(196,218), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,216)(58,217)(59,218)(60,219)(61,220)(62,221)(63,222)(64,223)(65,224)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,186)(86,187)(87,188)(88,189)(89,190)(90,191)(91,192)(92,193)(93,194)(94,195)(95,196)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,141)(139,142)(140,143), (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), (1,224,43,79)(2,209,44,64)(3,222,45,77)(4,207,46,62)(5,220,47,75)(6,205,48,60)(7,218,49,73)(8,203,50,58)(9,216,51,71)(10,201,52,84)(11,214,53,69)(12,199,54,82)(13,212,55,67)(14,197,56,80)(15,210,29,65)(16,223,30,78)(17,208,31,63)(18,221,32,76)(19,206,33,61)(20,219,34,74)(21,204,35,59)(22,217,36,72)(23,202,37,57)(24,215,38,70)(25,200,39,83)(26,213,40,68)(27,198,41,81)(28,211,42,66)(85,135,172,152)(86,120,173,165)(87,133,174,150)(88,118,175,163)(89,131,176,148)(90,116,177,161)(91,129,178,146)(92,114,179,159)(93,127,180,144)(94,140,181,157)(95,125,182,142)(96,138,183,155)(97,123,184,168)(98,136,185,153)(99,121,186,166)(100,134,187,151)(101,119,188,164)(102,132,189,149)(103,117,190,162)(104,130,191,147)(105,115,192,160)(106,128,193,145)(107,113,194,158)(108,126,195,143)(109,139,196,156)(110,124,169,141)(111,137,170,154)(112,122,171,167) );
G=PermutationGroup([[(2,134),(4,136),(6,138),(8,140),(10,114),(12,116),(14,118),(16,120),(18,122),(20,124),(22,126),(24,128),(26,130),(28,132),(30,165),(32,167),(34,141),(36,143),(38,145),(40,147),(42,149),(44,151),(46,153),(48,155),(50,157),(52,159),(54,161),(56,163),(57,180),(58,203),(59,182),(60,205),(61,184),(62,207),(63,186),(64,209),(65,188),(66,211),(67,190),(68,213),(69,192),(70,215),(71,194),(72,217),(73,196),(74,219),(75,170),(76,221),(77,172),(78,223),(79,174),(80,197),(81,176),(82,199),(83,178),(84,201),(85,222),(86,173),(87,224),(88,175),(89,198),(90,177),(91,200),(92,179),(93,202),(94,181),(95,204),(96,183),(97,206),(98,185),(99,208),(100,187),(101,210),(102,189),(103,212),(104,191),(105,214),(106,193),(107,216),(108,195),(109,218),(110,169),(111,220),(112,171)], [(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)], [(1,133),(2,134),(3,135),(4,136),(5,137),(6,138),(7,139),(8,140),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,164),(30,165),(31,166),(32,167),(33,168),(34,141),(35,142),(36,143),(37,144),(38,145),(39,146),(40,147),(41,148),(42,149),(43,150),(44,151),(45,152),(46,153),(47,154),(48,155),(49,156),(50,157),(51,158),(52,159),(53,160),(54,161),(55,162),(56,163),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,85),(78,86),(79,87),(80,88),(81,89),(82,90),(83,91),(84,92),(169,219),(170,220),(171,221),(172,222),(173,223),(174,224),(175,197),(176,198),(177,199),(178,200),(179,201),(180,202),(181,203),(182,204),(183,205),(184,206),(185,207),(186,208),(187,209),(188,210),(189,211),(190,212),(191,213),(192,214),(193,215),(194,216),(195,217),(196,218)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,216),(58,217),(59,218),(60,219),(61,220),(62,221),(63,222),(64,223),(65,224),(66,197),(67,198),(68,199),(69,200),(70,201),(71,202),(72,203),(73,204),(74,205),(75,206),(76,207),(77,208),(78,209),(79,210),(80,211),(81,212),(82,213),(83,214),(84,215),(85,186),(86,187),(87,188),(88,189),(89,190),(90,191),(91,192),(92,193),(93,194),(94,195),(95,196),(96,169),(97,170),(98,171),(99,172),(100,173),(101,174),(102,175),(103,176),(104,177),(105,178),(106,179),(107,180),(108,181),(109,182),(110,183),(111,184),(112,185),(113,144),(114,145),(115,146),(116,147),(117,148),(118,149),(119,150),(120,151),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(129,160),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,141),(139,142),(140,143)], [(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)], [(1,224,43,79),(2,209,44,64),(3,222,45,77),(4,207,46,62),(5,220,47,75),(6,205,48,60),(7,218,49,73),(8,203,50,58),(9,216,51,71),(10,201,52,84),(11,214,53,69),(12,199,54,82),(13,212,55,67),(14,197,56,80),(15,210,29,65),(16,223,30,78),(17,208,31,63),(18,221,32,76),(19,206,33,61),(20,219,34,74),(21,204,35,59),(22,217,36,72),(23,202,37,57),(24,215,38,70),(25,200,39,83),(26,213,40,68),(27,198,41,81),(28,211,42,66),(85,135,172,152),(86,120,173,165),(87,133,174,150),(88,118,175,163),(89,131,176,148),(90,116,177,161),(91,129,178,146),(92,114,179,159),(93,127,180,144),(94,140,181,157),(95,125,182,142),(96,138,183,155),(97,123,184,168),(98,136,185,153),(99,121,186,166),(100,134,187,151),(101,119,188,164),(102,132,189,149),(103,117,190,162),(104,130,191,147),(105,115,192,160),(106,128,193,145),(107,113,194,158),(108,126,195,143),(109,139,196,156),(110,124,169,141),(111,137,170,154),(112,122,171,167)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 |
| kernel | C24.4D14 | C14.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C23.D7 | C14×C22⋊C4 | C23.D7 | C2×Dic7 | C2×C28 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 3 | 8 | 6 | 3 | 12 | 12 | 12 | 3 | 9 |
Matrix representation of C24.4D14 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 23 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 24 |
| 0 | 0 | 0 | 0 | 28 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 28 | 6 | 0 | 0 |
| 0 | 19 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 12 |
| 0 | 0 | 0 | 0 | 5 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 17 | 14 | 0 | 0 |
| 0 | 25 | 12 | 0 | 0 |
| 0 | 0 | 0 | 6 | 16 |
| 0 | 0 | 0 | 14 | 23 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,23,28,0,0,0,0,0,1,0,0,0,0,24,28],[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],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[17,0,0,0,0,0,28,19,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,12,5],[28,0,0,0,0,0,17,25,0,0,0,14,12,0,0,0,0,0,6,14,0,0,0,16,23] >;
C24.4D14 in GAP, Magma, Sage, TeX
C_2^4._4D_{14} % in TeX
G:=Group("C2^4.4D14"); // GroupNames label
G:=SmallGroup(448,479);
// by ID
G=gap.SmallGroup(448,479);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,64,926,219,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=d*b=b*d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,b*c=c*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