metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.44D14, C23.9Dic14, C23.49(C4×D7), C22.94(D4×D7), (C22×Dic7)⋊7C4, (C22×C4).22D14, (C22×C14).60D4, (C22×C14).10Q8, C14.81(C4⋊D4), C7⋊2(C23.7Q8), C14.C42⋊9C2, C22⋊1(Dic7⋊C4), Dic7⋊3(C22⋊C4), (C2×Dic7).171D4, C23.35(C7⋊D4), C14.15(C22⋊Q8), (C23×Dic7).2C2, C2.1(Dic7⋊D4), (C23×C14).25C22, (C22×C28).21C22, C22.23(C2×Dic14), C23.275(C22×D7), C14.24(C42⋊C2), C22.41(D4⋊2D7), (C22×C14).317C23, C2.5(C22⋊Dic14), (C22×Dic7).34C22, C2.11(C23.11D14), (C2×C14)⋊1(C4⋊C4), C14.29(C2×C4⋊C4), (C2×Dic7⋊C4)⋊6C2, C2.5(C2×Dic7⋊C4), (C2×C22⋊C4).4D7, (C2×C14).30(C2×Q8), C2.27(D7×C22⋊C4), C22.121(C2×C4×D7), (C2×C14).429(C2×D4), (C14×C22⋊C4).5C2, C14.27(C2×C22⋊C4), C22.45(C2×C7⋊D4), (C2×C23.D7).4C2, (C22×C14).45(C2×C4), (C2×Dic7).92(C2×C4), (C2×C14).138(C4○D4), (C2×C14).103(C22×C4), SmallGroup(448,476)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.44D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce13 >
Subgroups: 916 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C22×C14, C22×C14, C23.7Q8, Dic7⋊C4, C23.D7, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×Dic7⋊C4, C2×C23.D7, C14×C22⋊C4, C23×Dic7, C24.44D14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, Dic14, C4×D7, C7⋊D4, C22×D7, C23.7Q8, Dic7⋊C4, C2×Dic14, C2×C4×D7, D4×D7, D4⋊2D7, C2×C7⋊D4, C23.11D14, C22⋊Dic14, D7×C22⋊C4, C2×Dic7⋊C4, Dic7⋊D4, C24.44D14
►On 224 points
Generators in S224
(1 102)(2 79)(3 104)(4 81)(5 106)(6 83)(7 108)(8 57)(9 110)(10 59)(11 112)(12 61)(13 86)(14 63)(15 88)(16 65)(17 90)(18 67)(19 92)(20 69)(21 94)(22 71)(23 96)(24 73)(25 98)(26 75)(27 100)(28 77)(29 129)(30 174)(31 131)(32 176)(33 133)(34 178)(35 135)(36 180)(37 137)(38 182)(39 139)(40 184)(41 113)(42 186)(43 115)(44 188)(45 117)(46 190)(47 119)(48 192)(49 121)(50 194)(51 123)(52 196)(53 125)(54 170)(55 127)(56 172)(58 204)(60 206)(62 208)(64 210)(66 212)(68 214)(70 216)(72 218)(74 220)(76 222)(78 224)(80 198)(82 200)(84 202)(85 207)(87 209)(89 211)(91 213)(93 215)(95 217)(97 219)(99 221)(101 223)(103 197)(105 199)(107 201)(109 203)(111 205)(114 163)(116 165)(118 167)(120 141)(122 143)(124 145)(126 147)(128 149)(130 151)(132 153)(134 155)(136 157)(138 159)(140 161)(142 193)(144 195)(146 169)(148 171)(150 173)(152 175)(154 177)(156 179)(158 181)(160 183)(162 185)(164 187)(166 189)(168 191)
(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 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 85)(27 86)(28 87)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 113)(56 114)(57 217)(58 218)(59 219)(60 220)(61 221)(62 222)(63 223)(64 224)(65 197)(66 198)(67 199)(68 200)(69 201)(70 202)(71 203)(72 204)(73 205)(74 206)(75 207)(76 208)(77 209)(78 210)(79 211)(80 212)(81 213)(82 214)(83 215)(84 216)(141 178)(142 179)(143 180)(144 181)(145 182)(146 183)(147 184)(148 185)(149 186)(150 187)(151 188)(152 189)(153 190)(154 191)(155 192)(156 193)(157 194)(158 195)(159 196)(160 169)(161 170)(162 171)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)
(1 224)(2 197)(3 198)(4 199)(5 200)(6 201)(7 202)(8 203)(9 204)(10 205)(11 206)(12 207)(13 208)(14 209)(15 210)(16 211)(17 212)(18 213)(19 214)(20 215)(21 216)(22 217)(23 218)(24 219)(25 220)(26 221)(27 222)(28 223)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 141)(49 142)(50 143)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 109)(58 110)(59 111)(60 112)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)(121 193)(122 194)(123 195)(124 196)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 181)(138 182)(139 183)(140 184)
(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)
(1 120 15 134)(2 47 16 33)(3 118 17 132)(4 45 18 31)(5 116 19 130)(6 43 20 29)(7 114 21 128)(8 41 22 55)(9 140 23 126)(10 39 24 53)(11 138 25 124)(12 37 26 51)(13 136 27 122)(14 35 28 49)(30 106 44 92)(32 104 46 90)(34 102 48 88)(36 100 50 86)(38 98 52 112)(40 96 54 110)(42 94 56 108)(57 185 71 171)(58 161 72 147)(59 183 73 169)(60 159 74 145)(61 181 75 195)(62 157 76 143)(63 179 77 193)(64 155 78 141)(65 177 79 191)(66 153 80 167)(67 175 81 189)(68 151 82 165)(69 173 83 187)(70 149 84 163)(85 137 99 123)(87 135 101 121)(89 133 103 119)(91 131 105 117)(93 129 107 115)(95 127 109 113)(97 125 111 139)(142 209 156 223)(144 207 158 221)(146 205 160 219)(148 203 162 217)(150 201 164 215)(152 199 166 213)(154 197 168 211)(170 204 184 218)(172 202 186 216)(174 200 188 214)(176 198 190 212)(178 224 192 210)(180 222 194 208)(182 220 196 206)
G:=sub<Sym(224)| (1,102)(2,79)(3,104)(4,81)(5,106)(6,83)(7,108)(8,57)(9,110)(10,59)(11,112)(12,61)(13,86)(14,63)(15,88)(16,65)(17,90)(18,67)(19,92)(20,69)(21,94)(22,71)(23,96)(24,73)(25,98)(26,75)(27,100)(28,77)(29,129)(30,174)(31,131)(32,176)(33,133)(34,178)(35,135)(36,180)(37,137)(38,182)(39,139)(40,184)(41,113)(42,186)(43,115)(44,188)(45,117)(46,190)(47,119)(48,192)(49,121)(50,194)(51,123)(52,196)(53,125)(54,170)(55,127)(56,172)(58,204)(60,206)(62,208)(64,210)(66,212)(68,214)(70,216)(72,218)(74,220)(76,222)(78,224)(80,198)(82,200)(84,202)(85,207)(87,209)(89,211)(91,213)(93,215)(95,217)(97,219)(99,221)(101,223)(103,197)(105,199)(107,201)(109,203)(111,205)(114,163)(116,165)(118,167)(120,141)(122,143)(124,145)(126,147)(128,149)(130,151)(132,153)(134,155)(136,157)(138,159)(140,161)(142,193)(144,195)(146,169)(148,171)(150,173)(152,175)(154,177)(156,179)(158,181)(160,183)(162,185)(164,187)(166,189)(168,191), (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), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,85)(27,86)(28,87)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,113)(56,114)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(65,197)(66,198)(67,199)(68,200)(69,201)(70,202)(71,203)(72,204)(73,205)(74,206)(75,207)(76,208)(77,209)(78,210)(79,211)(80,212)(81,213)(82,214)(83,215)(84,216)(141,178)(142,179)(143,180)(144,181)(145,182)(146,183)(147,184)(148,185)(149,186)(150,187)(151,188)(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)(159,196)(160,169)(161,170)(162,171)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177), (1,224)(2,197)(3,198)(4,199)(5,200)(6,201)(7,202)(8,203)(9,204)(10,205)(11,206)(12,207)(13,208)(14,209)(15,210)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,217)(23,218)(24,219)(25,220)(26,221)(27,222)(28,223)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,109)(58,110)(59,111)(60,112)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184), (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), (1,120,15,134)(2,47,16,33)(3,118,17,132)(4,45,18,31)(5,116,19,130)(6,43,20,29)(7,114,21,128)(8,41,22,55)(9,140,23,126)(10,39,24,53)(11,138,25,124)(12,37,26,51)(13,136,27,122)(14,35,28,49)(30,106,44,92)(32,104,46,90)(34,102,48,88)(36,100,50,86)(38,98,52,112)(40,96,54,110)(42,94,56,108)(57,185,71,171)(58,161,72,147)(59,183,73,169)(60,159,74,145)(61,181,75,195)(62,157,76,143)(63,179,77,193)(64,155,78,141)(65,177,79,191)(66,153,80,167)(67,175,81,189)(68,151,82,165)(69,173,83,187)(70,149,84,163)(85,137,99,123)(87,135,101,121)(89,133,103,119)(91,131,105,117)(93,129,107,115)(95,127,109,113)(97,125,111,139)(142,209,156,223)(144,207,158,221)(146,205,160,219)(148,203,162,217)(150,201,164,215)(152,199,166,213)(154,197,168,211)(170,204,184,218)(172,202,186,216)(174,200,188,214)(176,198,190,212)(178,224,192,210)(180,222,194,208)(182,220,196,206)>;
G:=Group( (1,102)(2,79)(3,104)(4,81)(5,106)(6,83)(7,108)(8,57)(9,110)(10,59)(11,112)(12,61)(13,86)(14,63)(15,88)(16,65)(17,90)(18,67)(19,92)(20,69)(21,94)(22,71)(23,96)(24,73)(25,98)(26,75)(27,100)(28,77)(29,129)(30,174)(31,131)(32,176)(33,133)(34,178)(35,135)(36,180)(37,137)(38,182)(39,139)(40,184)(41,113)(42,186)(43,115)(44,188)(45,117)(46,190)(47,119)(48,192)(49,121)(50,194)(51,123)(52,196)(53,125)(54,170)(55,127)(56,172)(58,204)(60,206)(62,208)(64,210)(66,212)(68,214)(70,216)(72,218)(74,220)(76,222)(78,224)(80,198)(82,200)(84,202)(85,207)(87,209)(89,211)(91,213)(93,215)(95,217)(97,219)(99,221)(101,223)(103,197)(105,199)(107,201)(109,203)(111,205)(114,163)(116,165)(118,167)(120,141)(122,143)(124,145)(126,147)(128,149)(130,151)(132,153)(134,155)(136,157)(138,159)(140,161)(142,193)(144,195)(146,169)(148,171)(150,173)(152,175)(154,177)(156,179)(158,181)(160,183)(162,185)(164,187)(166,189)(168,191), (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), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,85)(27,86)(28,87)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,113)(56,114)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(65,197)(66,198)(67,199)(68,200)(69,201)(70,202)(71,203)(72,204)(73,205)(74,206)(75,207)(76,208)(77,209)(78,210)(79,211)(80,212)(81,213)(82,214)(83,215)(84,216)(141,178)(142,179)(143,180)(144,181)(145,182)(146,183)(147,184)(148,185)(149,186)(150,187)(151,188)(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)(159,196)(160,169)(161,170)(162,171)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177), (1,224)(2,197)(3,198)(4,199)(5,200)(6,201)(7,202)(8,203)(9,204)(10,205)(11,206)(12,207)(13,208)(14,209)(15,210)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,217)(23,218)(24,219)(25,220)(26,221)(27,222)(28,223)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,109)(58,110)(59,111)(60,112)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184), (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), (1,120,15,134)(2,47,16,33)(3,118,17,132)(4,45,18,31)(5,116,19,130)(6,43,20,29)(7,114,21,128)(8,41,22,55)(9,140,23,126)(10,39,24,53)(11,138,25,124)(12,37,26,51)(13,136,27,122)(14,35,28,49)(30,106,44,92)(32,104,46,90)(34,102,48,88)(36,100,50,86)(38,98,52,112)(40,96,54,110)(42,94,56,108)(57,185,71,171)(58,161,72,147)(59,183,73,169)(60,159,74,145)(61,181,75,195)(62,157,76,143)(63,179,77,193)(64,155,78,141)(65,177,79,191)(66,153,80,167)(67,175,81,189)(68,151,82,165)(69,173,83,187)(70,149,84,163)(85,137,99,123)(87,135,101,121)(89,133,103,119)(91,131,105,117)(93,129,107,115)(95,127,109,113)(97,125,111,139)(142,209,156,223)(144,207,158,221)(146,205,160,219)(148,203,162,217)(150,201,164,215)(152,199,166,213)(154,197,168,211)(170,204,184,218)(172,202,186,216)(174,200,188,214)(176,198,190,212)(178,224,192,210)(180,222,194,208)(182,220,196,206) );
G=PermutationGroup([[(1,102),(2,79),(3,104),(4,81),(5,106),(6,83),(7,108),(8,57),(9,110),(10,59),(11,112),(12,61),(13,86),(14,63),(15,88),(16,65),(17,90),(18,67),(19,92),(20,69),(21,94),(22,71),(23,96),(24,73),(25,98),(26,75),(27,100),(28,77),(29,129),(30,174),(31,131),(32,176),(33,133),(34,178),(35,135),(36,180),(37,137),(38,182),(39,139),(40,184),(41,113),(42,186),(43,115),(44,188),(45,117),(46,190),(47,119),(48,192),(49,121),(50,194),(51,123),(52,196),(53,125),(54,170),(55,127),(56,172),(58,204),(60,206),(62,208),(64,210),(66,212),(68,214),(70,216),(72,218),(74,220),(76,222),(78,224),(80,198),(82,200),(84,202),(85,207),(87,209),(89,211),(91,213),(93,215),(95,217),(97,219),(99,221),(101,223),(103,197),(105,199),(107,201),(109,203),(111,205),(114,163),(116,165),(118,167),(120,141),(122,143),(124,145),(126,147),(128,149),(130,151),(132,153),(134,155),(136,157),(138,159),(140,161),(142,193),(144,195),(146,169),(148,171),(150,173),(152,175),(154,177),(156,179),(158,181),(160,183),(162,185),(164,187),(166,189),(168,191)], [(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)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,85),(27,86),(28,87),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,113),(56,114),(57,217),(58,218),(59,219),(60,220),(61,221),(62,222),(63,223),(64,224),(65,197),(66,198),(67,199),(68,200),(69,201),(70,202),(71,203),(72,204),(73,205),(74,206),(75,207),(76,208),(77,209),(78,210),(79,211),(80,212),(81,213),(82,214),(83,215),(84,216),(141,178),(142,179),(143,180),(144,181),(145,182),(146,183),(147,184),(148,185),(149,186),(150,187),(151,188),(152,189),(153,190),(154,191),(155,192),(156,193),(157,194),(158,195),(159,196),(160,169),(161,170),(162,171),(163,172),(164,173),(165,174),(166,175),(167,176),(168,177)], [(1,224),(2,197),(3,198),(4,199),(5,200),(6,201),(7,202),(8,203),(9,204),(10,205),(11,206),(12,207),(13,208),(14,209),(15,210),(16,211),(17,212),(18,213),(19,214),(20,215),(21,216),(22,217),(23,218),(24,219),(25,220),(26,221),(27,222),(28,223),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,157),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,141),(49,142),(50,143),(51,144),(52,145),(53,146),(54,147),(55,148),(56,149),(57,109),(58,110),(59,111),(60,112),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192),(121,193),(122,194),(123,195),(124,196),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176),(133,177),(134,178),(135,179),(136,180),(137,181),(138,182),(139,183),(140,184)], [(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)], [(1,120,15,134),(2,47,16,33),(3,118,17,132),(4,45,18,31),(5,116,19,130),(6,43,20,29),(7,114,21,128),(8,41,22,55),(9,140,23,126),(10,39,24,53),(11,138,25,124),(12,37,26,51),(13,136,27,122),(14,35,28,49),(30,106,44,92),(32,104,46,90),(34,102,48,88),(36,100,50,86),(38,98,52,112),(40,96,54,110),(42,94,56,108),(57,185,71,171),(58,161,72,147),(59,183,73,169),(60,159,74,145),(61,181,75,195),(62,157,76,143),(63,179,77,193),(64,155,78,141),(65,177,79,191),(66,153,80,167),(67,175,81,189),(68,151,82,165),(69,173,83,187),(70,149,84,163),(85,137,99,123),(87,135,101,121),(89,133,103,119),(91,131,105,117),(93,129,107,115),(95,127,109,113),(97,125,111,139),(142,209,156,223),(144,207,158,221),(146,205,160,219),(148,203,162,217),(150,201,164,215),(152,199,166,213),(154,197,168,211),(170,204,184,218),(172,202,186,216),(174,200,188,214),(176,198,190,212),(178,224,192,210),(180,222,194,208),(182,220,196,206)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D7 | C4○D4 | D14 | D14 | Dic14 | C4×D7 | C7⋊D4 | D4×D7 | D4⋊2D7 |
| kernel | C24.44D14 | C14.C42 | C2×Dic7⋊C4 | C2×C23.D7 | C14×C22⋊C4 | C23×Dic7 | C22×Dic7 | C2×Dic7 | C22×C14 | C22×C14 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 2 | 2 | 3 | 4 | 6 | 3 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C24.44D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 23 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 24 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 27 | 0 | 0 |
| 0 | 0 | 18 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 27 |
| 0 | 0 | 0 | 0 | 4 | 6 |
| 4 | 28 | 0 | 0 | 0 | 0 |
| 15 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 13 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 5 |
| 0 | 0 | 0 | 0 | 7 | 14 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,23,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[27,24,0,0,0,0,1,3,0,0,0,0,0,0,8,18,0,0,0,0,27,21,0,0,0,0,0,0,23,4,0,0,0,0,27,6],[4,15,0,0,0,0,28,25,0,0,0,0,0,0,9,13,0,0,0,0,16,20,0,0,0,0,0,0,15,7,0,0,0,0,5,14] >;
C24.44D14 in GAP, Magma, Sage, TeX
C_2^4._{44}D_{14} % in TeX
G:=Group("C2^4.44D14"); // GroupNames label
G:=SmallGroup(448,476);
// by ID
G=gap.SmallGroup(448,476);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,387,58,18822]);
// 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=f*a*f^-1=a*d=d*a,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=c*e^13>;
// generators/relations