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G = C2×C23.D13order 416 = 25·13

Direct product of C2 and C23.D13

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

Aliases: C2×C23.D13, C24.D13, C232Dic13, C23.32D26, (C22×C26)⋊7C4, C26.62(C2×D4), (C2×C26).44D4, C263(C22⋊C4), (C23×C26).2C2, C26.41(C22×C4), (C2×C26).60C23, C222(C2×Dic13), (C2×Dic13)⋊7C22, (C22×Dic13)⋊7C2, C2.9(C22×Dic13), C22.25(C13⋊D4), (C22×C26).41C22, C22.27(C22×D13), C134(C2×C22⋊C4), (C2×C26)⋊11(C2×C4), C2.4(C2×C13⋊D4), SmallGroup(416,173)

Series: Derived Chief Lower central Upper central

C1C26 — C2×C23.D13
C1C13C26C2×C26C2×Dic13C22×Dic13 — C2×C23.D13
C13C26 — C2×C23.D13
C1C23C24

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 [×6], C2 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C2×C4 [×8], C23, C23 [×6], C23 [×4], C13, C22⋊C4 [×4], C22×C4 [×2], C24, C26, C26 [×6], C26 [×4], C2×C22⋊C4, Dic13 [×4], C2×C26, C2×C26 [×10], C2×C26 [×12], C2×Dic13 [×4], C2×Dic13 [×4], C22×C26, C22×C26 [×6], C22×C26 [×4], C23.D13 [×4], C22×Dic13 [×2], C23×C26, C2×C23.D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D13, C2×C22⋊C4, Dic13 [×4], D26 [×3], C2×Dic13 [×6], C13⋊D4 [×4], C22×D13, C23.D13 [×4], C22×Dic13, C2×C13⋊D4 [×2], 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)(105 201)(106 202)(107 203)(108 204)(109 205)(110 206)(111 207)(112 208)(113 196)(114 197)(115 198)(116 199)(117 200)(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)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 181)(138 182)(139 170)(140 171)(141 172)(142 173)(143 174)(144 160)(145 161)(146 162)(147 163)(148 164)(149 165)(150 166)(151 167)(152 168)(153 169)(154 157)(155 158)(156 159)
(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)(131 150)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 144)(139 145)(140 146)(141 147)(142 148)(143 149)(157 179)(158 180)(159 181)(160 182)(161 170)(162 171)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(169 178)(183 206)(184 207)(185 208)(186 196)(187 197)(188 198)(189 199)(190 200)(191 201)(192 202)(193 203)(194 204)(195 205)
(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)(105 140)(106 141)(107 142)(108 143)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 144)(130 145)(157 186)(158 187)(159 188)(160 189)(161 190)(162 191)(163 192)(164 193)(165 194)(166 195)(167 183)(168 184)(169 185)(170 200)(171 201)(172 202)(173 203)(174 204)(175 205)(176 206)(177 207)(178 208)(179 196)(180 197)(181 198)(182 199)
(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)(105 118)(106 119)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(131 150)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 144)(139 145)(140 146)(141 147)(142 148)(143 149)(157 179)(158 180)(159 181)(160 182)(161 170)(162 171)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(169 178)(183 206)(184 207)(185 208)(186 196)(187 197)(188 198)(189 199)(190 200)(191 201)(192 202)(193 203)(194 204)(195 205)
(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 27 116)(2 137 28 115)(3 136 29 114)(4 135 30 113)(5 134 31 112)(6 133 32 111)(7 132 33 110)(8 131 34 109)(9 143 35 108)(10 142 36 107)(11 141 37 106)(12 140 38 105)(13 139 39 117)(14 144 40 129)(15 156 41 128)(16 155 42 127)(17 154 43 126)(18 153 44 125)(19 152 45 124)(20 151 46 123)(21 150 47 122)(22 149 48 121)(23 148 49 120)(24 147 50 119)(25 146 51 118)(26 145 52 130)(53 189 79 160)(54 188 80 159)(55 187 81 158)(56 186 82 157)(57 185 83 169)(58 184 84 168)(59 183 85 167)(60 195 86 166)(61 194 87 165)(62 193 88 164)(63 192 89 163)(64 191 90 162)(65 190 91 161)(66 199 92 182)(67 198 93 181)(68 197 94 180)(69 196 95 179)(70 208 96 178)(71 207 97 177)(72 206 98 176)(73 205 99 175)(74 204 100 174)(75 203 101 173)(76 202 102 172)(77 201 103 171)(78 200 104 170)

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

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

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

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

׿
×
𝔽