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G = C2×C23.D13  order 416 = 25·13

Direct product of C2 and C23.D13

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

Aliases: C2×C23.D13, C24.D13, C23⋊2Dic13, C23.32D26, (C22×C26)⋊7C4, C26.62(C2×D4), (C2×C26).44D4, C26⋊3(C22⋊C4), (C23×C26).2C2, C26.41(C22×C4), (C2×C26).60C23, C22⋊2(C2×Dic13), (C2×Dic13)⋊7C22, (C22×Dic13)⋊7C2, C2.9(C22×Dic13), C22.25(C13⋊D4), (C22×C26).41C22, C22.27(C22×D13), C13⋊4(C2×C22⋊C4), (C2×C26)⋊11(C2×C4), C2.4(C2×C13⋊D4), SmallGroup(416,173)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C26 — C2×C23.D13
Chief series
C1 — C13 — C26 — C2×C26 — C2×Dic13 — C22×Dic13 — C2×C23.D13
Lower central
C13 — C26 — C2×C23.D13
Upper central
C1 — C23 — C24

Generators and relations for C2×C23.D13
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e13=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 512 in 132 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C23, C23, C23, C13, C22⋊C4, C22×C4, C24, C26, C26, C26, C2×C22⋊C4, Dic13, C2×C26, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C22×C26, C22×C26, C22×C26, C23.D13, C22×Dic13, C23×C26, C2×C23.D13
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, D13, C2×C22⋊C4, Dic13, D26, C2×Dic13, C13⋊D4, C22×D13, C23.D13, C22×Dic13, C2×C13⋊D4, C2×C23.D13

Smallest permutation representation of C2×C23.D13
►On 208 points
Generators in S208
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)(51 64)(52 65)(105 207)(106 208)(107 196)(108 197)(109 198)(110 199)(111 200)(112 201)(113 202)(114 203)(115 204)(116 205)(117 206)(118 191)(119 192)(120 193)(121 194)(122 195)(123 183)(124 184)(125 185)(126 186)(127 187)(128 188)(129 189)(130 190)(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)(144 161)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)(151 168)(152 169)(153 157)(154 158)(155 159)(156 160)

(105 128)(106 129)(107 130)(108 118)(109 119)(110 120)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)(157 174)(158 175)(159 176)(160 177)(161 178)(162 179)(163 180)(164 181)(165 182)(166 170)(167 171)(168 172)(169 173)(183 202)(184 203)(185 204)(186 205)(187 206)(188 207)(189 208)(190 196)(191 197)(192 198)(193 199)(194 200)(195 201)

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

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

(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)

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

116 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H13A···13F26A···26CL
order12···222224···413···1326···26
size11···1222226···262···22···2

116 irreducible representations

dim1111122222
type++++++-+
imageC1C2C2C2C4D4D13Dic13D26C13⋊D4
kernelC2×C23.D13C23.D13C22×Dic13C23×C26C22×C26C2×C26C24C23C23C22
# reps1421846241848

Matrix representation of C2×C23.D13 ►in GL4(𝔽53) generated by

52000
05200
00520
00052
,
52000
05200
0010
002652
,
52000
0100
0010
0001
,
1000
0100
00520
00052
,
1000
0100
00440
001447
,
30000
0100
003118
003522
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,1,26,0,0,0,52],[52,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,44,14,0,0,0,47],[30,0,0,0,0,1,0,0,0,0,31,35,0,0,18,22] >;
 

C2×C23.D13 in GAP, Magma, Sage, TeX 

C_2\times C_2^3.D_{13}
 
% in TeX
 
G:=Group("C2xC2^3.D13");
 
// GroupNames label
 
G:=SmallGroup(416,173);
 
// by ID
 
G=gap.SmallGroup(416,173);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,13829]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^13=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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^-1>;
 
// generators/relations
 

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×
⋊
ℤ
𝔽
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