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G = C24×D13  order 416 = 25·13

Direct product of C24 and D13

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C24×D13, C13⋊C25, C26⋊C24, (C23×C26)⋊5C2, (C2×C26)⋊4C23, (C22×C26)⋊8C22, SmallGroup(416,234)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C13 — C24×D13
Chief series
C1 — C13 — D13 — D26 — C22×D13 — C23×D13 — C24×D13
Lower central
C13 — C24×D13
Upper central
C1 — C24

Generators and relations for C24×D13
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e13=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 4432 in 748 conjugacy classes, 441 normal (5 characteristic)
C1, C2, C2, C22, C22, C23, C23, C13, C24, C24, D13, C26, C25, D26, C2×C26, C22×D13, C22×C26, C23×D13, C23×C26, C24×D13
Quotients: C1, C2, C22, C23, C24, D13, C25, D26, C22×D13, C23×D13, C24×D13

Smallest permutation representation of C24×D13
►On 208 points
Generators in S208
(1 196)(2 197)(3 198)(4 199)(5 200)(6 201)(7 202)(8 203)(9 204)(10 205)(11 206)(12 207)(13 208)(14 188)(15 189)(16 190)(17 191)(18 192)(19 193)(20 194)(21 195)(22 183)(23 184)(24 185)(25 186)(26 187)(27 170)(28 171)(29 172)(30 173)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 180)(38 181)(39 182)(40 168)(41 169)(42 157)(43 158)(44 159)(45 160)(46 161)(47 162)(48 163)(49 164)(50 165)(51 166)(52 167)(53 148)(54 149)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 156)(62 144)(63 145)(64 146)(65 147)(66 139)(67 140)(68 141)(69 142)(70 143)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 118)(89 119)(90 120)(91 121)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)(101 114)(102 115)(103 116)(104 117)

(1 101)(2 102)(3 103)(4 104)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 88)(15 89)(16 90)(17 91)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 76)(28 77)(29 78)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 53)(48 54)(49 55)(50 56)(51 57)(52 58)(105 200)(106 201)(107 202)(108 203)(109 204)(110 205)(111 206)(112 207)(113 208)(114 196)(115 197)(116 198)(117 199)(118 188)(119 189)(120 190)(121 191)(122 192)(123 193)(124 194)(125 195)(126 183)(127 184)(128 185)(129 186)(130 187)(131 178)(132 179)(133 180)(134 181)(135 182)(136 170)(137 171)(138 172)(139 173)(140 174)(141 175)(142 176)(143 177)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(155 169)(156 157)

(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 27)(12 28)(13 29)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(53 89)(54 90)(55 91)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 101)(67 102)(68 103)(69 104)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)(105 143)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 144)(129 145)(130 146)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 193)(167 194)(168 195)(169 183)(170 206)(171 207)(172 208)(173 196)(174 197)(175 198)(176 199)(177 200)(178 201)(179 202)(180 203)(181 204)(182 205)

(1 23)(2 24)(3 25)(4 26)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(27 52)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 66)(62 67)(63 68)(64 69)(65 70)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(86 103)(87 104)(88 92)(89 93)(90 94)(91 95)(105 118)(106 119)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 144)(141 145)(142 146)(143 147)(157 173)(158 174)(159 175)(160 176)(161 177)(162 178)(163 179)(164 180)(165 181)(166 182)(167 170)(168 171)(169 172)(183 208)(184 196)(185 197)(186 198)(187 199)(188 200)(189 201)(190 202)(191 203)(192 204)(193 205)(194 206)(195 207)

(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)

(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 143)(10 142)(11 141)(12 140)(13 139)(14 151)(15 150)(16 149)(17 148)(18 147)(19 146)(20 145)(21 144)(22 156)(23 155)(24 154)(25 153)(26 152)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 117)(40 128)(41 127)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 130)(52 129)(53 191)(54 190)(55 189)(56 188)(57 187)(58 186)(59 185)(60 184)(61 183)(62 195)(63 194)(64 193)(65 192)(66 208)(67 207)(68 206)(69 205)(70 204)(71 203)(72 202)(73 201)(74 200)(75 199)(76 198)(77 197)(78 196)(79 161)(80 160)(81 159)(82 158)(83 157)(84 169)(85 168)(86 167)(87 166)(88 165)(89 164)(90 163)(91 162)(92 181)(93 180)(94 179)(95 178)(96 177)(97 176)(98 175)(99 174)(100 173)(101 172)(102 171)(103 170)(104 182)
 
G:=sub<Sym(208)| (1,196)(2,197)(3,198)(4,199)(5,200)(6,201)(7,202)(8,203)(9,204)(10,205)(11,206)(12,207)(13,208)(14,188)(15,189)(16,190)(17,191)(18,192)(19,193)(20,194)(21,195)(22,183)(23,184)(24,185)(25,186)(26,187)(27,170)(28,171)(29,172)(30,173)(31,174)(32,175)(33,176)(34,177)(35,178)(36,179)(37,180)(38,181)(39,182)(40,168)(41,169)(42,157)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,144)(63,145)(64,146)(65,147)(66,139)(67,140)(68,141)(69,142)(70,143)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,118)(89,119)(90,120)(91,121)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117), (1,101)(2,102)(3,103)(4,104)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,88)(15,89)(16,90)(17,91)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,76)(28,77)(29,78)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(113,208)(114,196)(115,197)(116,198)(117,199)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,183)(127,184)(128,185)(129,186)(130,187)(131,178)(132,179)(133,180)(134,181)(135,182)(136,170)(137,171)(138,172)(139,173)(140,174)(141,175)(142,176)(143,177)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(155,169)(156,157), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,27)(12,28)(13,29)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(53,89)(54,90)(55,91)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,101)(67,102)(68,103)(69,104)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(105,143)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,144)(129,145)(130,146)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,183)(170,206)(171,207)(172,208)(173,196)(174,197)(175,198)(176,199)(177,200)(178,201)(179,202)(180,203)(181,204)(182,205), (1,23)(2,24)(3,25)(4,26)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(27,52)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,66)(62,67)(63,68)(64,69)(65,70)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(86,103)(87,104)(88,92)(89,93)(90,94)(91,95)(105,118)(106,119)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,144)(141,145)(142,146)(143,147)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,170)(168,171)(169,172)(183,208)(184,196)(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)(192,204)(193,205)(194,206)(195,207), (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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,143)(10,142)(11,141)(12,140)(13,139)(14,151)(15,150)(16,149)(17,148)(18,147)(19,146)(20,145)(21,144)(22,156)(23,155)(24,154)(25,153)(26,152)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,117)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,130)(52,129)(53,191)(54,190)(55,189)(56,188)(57,187)(58,186)(59,185)(60,184)(61,183)(62,195)(63,194)(64,193)(65,192)(66,208)(67,207)(68,206)(69,205)(70,204)(71,203)(72,202)(73,201)(74,200)(75,199)(76,198)(77,197)(78,196)(79,161)(80,160)(81,159)(82,158)(83,157)(84,169)(85,168)(86,167)(87,166)(88,165)(89,164)(90,163)(91,162)(92,181)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,182)>;
 
G:=Group( (1,196)(2,197)(3,198)(4,199)(5,200)(6,201)(7,202)(8,203)(9,204)(10,205)(11,206)(12,207)(13,208)(14,188)(15,189)(16,190)(17,191)(18,192)(19,193)(20,194)(21,195)(22,183)(23,184)(24,185)(25,186)(26,187)(27,170)(28,171)(29,172)(30,173)(31,174)(32,175)(33,176)(34,177)(35,178)(36,179)(37,180)(38,181)(39,182)(40,168)(41,169)(42,157)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,144)(63,145)(64,146)(65,147)(66,139)(67,140)(68,141)(69,142)(70,143)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,118)(89,119)(90,120)(91,121)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117), (1,101)(2,102)(3,103)(4,104)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,88)(15,89)(16,90)(17,91)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,76)(28,77)(29,78)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(113,208)(114,196)(115,197)(116,198)(117,199)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,183)(127,184)(128,185)(129,186)(130,187)(131,178)(132,179)(133,180)(134,181)(135,182)(136,170)(137,171)(138,172)(139,173)(140,174)(141,175)(142,176)(143,177)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(155,169)(156,157), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,27)(12,28)(13,29)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(53,89)(54,90)(55,91)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,101)(67,102)(68,103)(69,104)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(105,143)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,144)(129,145)(130,146)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,183)(170,206)(171,207)(172,208)(173,196)(174,197)(175,198)(176,199)(177,200)(178,201)(179,202)(180,203)(181,204)(182,205), (1,23)(2,24)(3,25)(4,26)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(27,52)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,66)(62,67)(63,68)(64,69)(65,70)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(86,103)(87,104)(88,92)(89,93)(90,94)(91,95)(105,118)(106,119)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,126)(114,127)(115,128)(116,129)(117,130)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,144)(141,145)(142,146)(143,147)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,170)(168,171)(169,172)(183,208)(184,196)(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)(192,204)(193,205)(194,206)(195,207), (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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,143)(10,142)(11,141)(12,140)(13,139)(14,151)(15,150)(16,149)(17,148)(18,147)(19,146)(20,145)(21,144)(22,156)(23,155)(24,154)(25,153)(26,152)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,117)(40,128)(41,127)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,130)(52,129)(53,191)(54,190)(55,189)(56,188)(57,187)(58,186)(59,185)(60,184)(61,183)(62,195)(63,194)(64,193)(65,192)(66,208)(67,207)(68,206)(69,205)(70,204)(71,203)(72,202)(73,201)(74,200)(75,199)(76,198)(77,197)(78,196)(79,161)(80,160)(81,159)(82,158)(83,157)(84,169)(85,168)(86,167)(87,166)(88,165)(89,164)(90,163)(91,162)(92,181)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,182) );
 
G=PermutationGroup([[(1,196),(2,197),(3,198),(4,199),(5,200),(6,201),(7,202),(8,203),(9,204),(10,205),(11,206),(12,207),(13,208),(14,188),(15,189),(16,190),(17,191),(18,192),(19,193),(20,194),(21,195),(22,183),(23,184),(24,185),(25,186),(26,187),(27,170),(28,171),(29,172),(30,173),(31,174),(32,175),(33,176),(34,177),(35,178),(36,179),(37,180),(38,181),(39,182),(40,168),(41,169),(42,157),(43,158),(44,159),(45,160),(46,161),(47,162),(48,163),(49,164),(50,165),(51,166),(52,167),(53,148),(54,149),(55,150),(56,151),(57,152),(58,153),(59,154),(60,155),(61,156),(62,144),(63,145),(64,146),(65,147),(66,139),(67,140),(68,141),(69,142),(70,143),(71,131),(72,132),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,122),(80,123),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,118),(89,119),(90,120),(91,121),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,111),(99,112),(100,113),(101,114),(102,115),(103,116),(104,117)], [(1,101),(2,102),(3,103),(4,104),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,88),(15,89),(16,90),(17,91),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,76),(28,77),(29,78),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,53),(48,54),(49,55),(50,56),(51,57),(52,58),(105,200),(106,201),(107,202),(108,203),(109,204),(110,205),(111,206),(112,207),(113,208),(114,196),(115,197),(116,198),(117,199),(118,188),(119,189),(120,190),(121,191),(122,192),(123,193),(124,194),(125,195),(126,183),(127,184),(128,185),(129,186),(130,187),(131,178),(132,179),(133,180),(134,181),(135,182),(136,170),(137,171),(138,172),(139,173),(140,174),(141,175),(142,176),(143,177),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(155,169),(156,157)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,27),(12,28),(13,29),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(53,89),(54,90),(55,91),(56,79),(57,80),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,101),(67,102),(68,103),(69,104),(70,92),(71,93),(72,94),(73,95),(74,96),(75,97),(76,98),(77,99),(78,100),(105,143),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,147),(119,148),(120,149),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,144),(129,145),(130,146),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(166,193),(167,194),(168,195),(169,183),(170,206),(171,207),(172,208),(173,196),(174,197),(175,198),(176,199),(177,200),(178,201),(179,202),(180,203),(181,204),(182,205)], [(1,23),(2,24),(3,25),(4,26),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(13,22),(27,52),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,66),(62,67),(63,68),(64,69),(65,70),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102),(86,103),(87,104),(88,92),(89,93),(90,94),(91,95),(105,118),(106,119),(107,120),(108,121),(109,122),(110,123),(111,124),(112,125),(113,126),(114,127),(115,128),(116,129),(117,130),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,144),(141,145),(142,146),(143,147),(157,173),(158,174),(159,175),(160,176),(161,177),(162,178),(163,179),(164,180),(165,181),(166,182),(167,170),(168,171),(169,172),(183,208),(184,196),(185,197),(186,198),(187,199),(188,200),(189,201),(190,202),(191,203),(192,204),(193,205),(194,206),(195,207)], [(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)], [(1,138),(2,137),(3,136),(4,135),(5,134),(6,133),(7,132),(8,131),(9,143),(10,142),(11,141),(12,140),(13,139),(14,151),(15,150),(16,149),(17,148),(18,147),(19,146),(20,145),(21,144),(22,156),(23,155),(24,154),(25,153),(26,152),(27,116),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,117),(40,128),(41,127),(42,126),(43,125),(44,124),(45,123),(46,122),(47,121),(48,120),(49,119),(50,118),(51,130),(52,129),(53,191),(54,190),(55,189),(56,188),(57,187),(58,186),(59,185),(60,184),(61,183),(62,195),(63,194),(64,193),(65,192),(66,208),(67,207),(68,206),(69,205),(70,204),(71,203),(72,202),(73,201),(74,200),(75,199),(76,198),(77,197),(78,196),(79,161),(80,160),(81,159),(82,158),(83,157),(84,169),(85,168),(86,167),(87,166),(88,165),(89,164),(90,163),(91,162),(92,181),(93,180),(94,179),(95,178),(96,177),(97,176),(98,175),(99,174),(100,173),(101,172),(102,171),(103,170),(104,182)]])
 

128 conjugacy classes

class 1 2A···2O2P···2AE13A···13F26A···26CL
order12···22···213···1326···26
size11···113···132···22···2

128 irreducible representations

dim11122
type+++++
imageC1C2C2D13D26
kernelC24×D13C23×D13C23×C26C24C23
# reps1301690

Matrix representation of C24×D13 ►in GL5(𝔽53)

520000
01000
00100
00010
00001
,
520000
01000
00100
000520
000052
,
520000
052000
00100
00010
00001
,
520000
052000
005200
000520
000052
,
10000
01000
00100
000431
0003118
,
10000
01000
005200
0003745
0001216

G:=sub<GL(5,GF(53))| [52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,52],[52,0,0,0,0,0,52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[52,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,43,31,0,0,0,1,18],[1,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,37,12,0,0,0,45,16] >;
 

C24×D13 in GAP, Magma, Sage, TeX 

C_2^4\times D_{13}
 
% in TeX
 
G:=Group("C2^4xD13");
 
// GroupNames label
 
G:=SmallGroup(416,234);
 
// by ID
 
G=gap.SmallGroup(416,234);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,13829]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^13=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*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=e^-1>;
 
// generators/relations
 

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