direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C13⋊2C8, C26⋊2C8, C52.6C4, C4.14D26, C4.3Dic13, C52.14C22, C22.2Dic13, C13⋊4(C2×C8), (C2×C52).6C2, (C2×C26).4C4, (C2×C4).5D13, C26.13(C2×C4), C2.1(C2×Dic13), SmallGroup(208,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C2×C13⋊2C8 |
Generators and relations for C2×C13⋊2C8
G = < a,b,c | a2=b13=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C13⋊2C8
►Regular action on 208 points
Generators in S208
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(105 166)(106 167)(107 168)(108 169)(109 157)(110 158)(111 159)(112 160)(113 161)(114 162)(115 163)(116 164)(117 165)(118 174)(119 175)(120 176)(121 177)(122 178)(123 179)(124 180)(125 181)(126 182)(127 170)(128 171)(129 172)(130 173)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 191)(139 192)(140 193)(141 194)(142 195)(143 183)(144 200)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 196)(154 197)(155 198)(156 199)
(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)
(1 200 40 175 14 187 27 163)(2 199 41 174 15 186 28 162)(3 198 42 173 16 185 29 161)(4 197 43 172 17 184 30 160)(5 196 44 171 18 183 31 159)(6 208 45 170 19 195 32 158)(7 207 46 182 20 194 33 157)(8 206 47 181 21 193 34 169)(9 205 48 180 22 192 35 168)(10 204 49 179 23 191 36 167)(11 203 50 178 24 190 37 166)(12 202 51 177 25 189 38 165)(13 201 52 176 26 188 39 164)(53 144 92 119 66 134 79 115)(54 156 93 118 67 133 80 114)(55 155 94 130 68 132 81 113)(56 154 95 129 69 131 82 112)(57 153 96 128 70 143 83 111)(58 152 97 127 71 142 84 110)(59 151 98 126 72 141 85 109)(60 150 99 125 73 140 86 108)(61 149 100 124 74 139 87 107)(62 148 101 123 75 138 88 106)(63 147 102 122 76 137 89 105)(64 146 103 121 77 136 90 117)(65 145 104 120 78 135 91 116)
G:=sub<Sym(208)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(105,166)(106,167)(107,168)(108,169)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,174)(119,175)(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,170)(128,171)(129,172)(130,173)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,183)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,196)(154,197)(155,198)(156,199), (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), (1,200,40,175,14,187,27,163)(2,199,41,174,15,186,28,162)(3,198,42,173,16,185,29,161)(4,197,43,172,17,184,30,160)(5,196,44,171,18,183,31,159)(6,208,45,170,19,195,32,158)(7,207,46,182,20,194,33,157)(8,206,47,181,21,193,34,169)(9,205,48,180,22,192,35,168)(10,204,49,179,23,191,36,167)(11,203,50,178,24,190,37,166)(12,202,51,177,25,189,38,165)(13,201,52,176,26,188,39,164)(53,144,92,119,66,134,79,115)(54,156,93,118,67,133,80,114)(55,155,94,130,68,132,81,113)(56,154,95,129,69,131,82,112)(57,153,96,128,70,143,83,111)(58,152,97,127,71,142,84,110)(59,151,98,126,72,141,85,109)(60,150,99,125,73,140,86,108)(61,149,100,124,74,139,87,107)(62,148,101,123,75,138,88,106)(63,147,102,122,76,137,89,105)(64,146,103,121,77,136,90,117)(65,145,104,120,78,135,91,116)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(105,166)(106,167)(107,168)(108,169)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,174)(119,175)(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,170)(128,171)(129,172)(130,173)(131,184)(132,185)(133,186)(134,187)(135,188)(136,189)(137,190)(138,191)(139,192)(140,193)(141,194)(142,195)(143,183)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,196)(154,197)(155,198)(156,199), (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), (1,200,40,175,14,187,27,163)(2,199,41,174,15,186,28,162)(3,198,42,173,16,185,29,161)(4,197,43,172,17,184,30,160)(5,196,44,171,18,183,31,159)(6,208,45,170,19,195,32,158)(7,207,46,182,20,194,33,157)(8,206,47,181,21,193,34,169)(9,205,48,180,22,192,35,168)(10,204,49,179,23,191,36,167)(11,203,50,178,24,190,37,166)(12,202,51,177,25,189,38,165)(13,201,52,176,26,188,39,164)(53,144,92,119,66,134,79,115)(54,156,93,118,67,133,80,114)(55,155,94,130,68,132,81,113)(56,154,95,129,69,131,82,112)(57,153,96,128,70,143,83,111)(58,152,97,127,71,142,84,110)(59,151,98,126,72,141,85,109)(60,150,99,125,73,140,86,108)(61,149,100,124,74,139,87,107)(62,148,101,123,75,138,88,106)(63,147,102,122,76,137,89,105)(64,146,103,121,77,136,90,117)(65,145,104,120,78,135,91,116) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(105,166),(106,167),(107,168),(108,169),(109,157),(110,158),(111,159),(112,160),(113,161),(114,162),(115,163),(116,164),(117,165),(118,174),(119,175),(120,176),(121,177),(122,178),(123,179),(124,180),(125,181),(126,182),(127,170),(128,171),(129,172),(130,173),(131,184),(132,185),(133,186),(134,187),(135,188),(136,189),(137,190),(138,191),(139,192),(140,193),(141,194),(142,195),(143,183),(144,200),(145,201),(146,202),(147,203),(148,204),(149,205),(150,206),(151,207),(152,208),(153,196),(154,197),(155,198),(156,199)], [(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)], [(1,200,40,175,14,187,27,163),(2,199,41,174,15,186,28,162),(3,198,42,173,16,185,29,161),(4,197,43,172,17,184,30,160),(5,196,44,171,18,183,31,159),(6,208,45,170,19,195,32,158),(7,207,46,182,20,194,33,157),(8,206,47,181,21,193,34,169),(9,205,48,180,22,192,35,168),(10,204,49,179,23,191,36,167),(11,203,50,178,24,190,37,166),(12,202,51,177,25,189,38,165),(13,201,52,176,26,188,39,164),(53,144,92,119,66,134,79,115),(54,156,93,118,67,133,80,114),(55,155,94,130,68,132,81,113),(56,154,95,129,69,131,82,112),(57,153,96,128,70,143,83,111),(58,152,97,127,71,142,84,110),(59,151,98,126,72,141,85,109),(60,150,99,125,73,140,86,108),(61,149,100,124,74,139,87,107),(62,148,101,123,75,138,88,106),(63,147,102,122,76,137,89,105),(64,146,103,121,77,136,90,117),(65,145,104,120,78,135,91,116)]])
C2×C13⋊2C8 is a maximal subgroup of
C26.7C42 C52⋊3C8 C26.D8 C52.Q8 D52⋊6C4 C26.Q16 C8×Dic13 C52.8Q8 C104⋊8C4 D26⋊1C8 C52.53D4 C52.55D4 D4⋊Dic13 Q8⋊Dic13 C52.C8 C2×C8×D13 D52.2C4 D4.Dic13 C52.C23
C2×C13⋊2C8 is a maximal quotient of
C52⋊3C8 C52.4C8 C52.55D4
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D13 | Dic13 | D26 | Dic13 | C13⋊2C8 |
| kernel | C2×C13⋊2C8 | C13⋊2C8 | C2×C52 | C52 | C2×C26 | C26 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 6 | 6 | 6 | 6 | 24 |
Matrix representation of C2×C13⋊2C8 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 312 |
| 1 | 0 | 0 |
| 0 | 78 | 312 |
| 0 | 79 | 312 |
| 288 | 0 | 0 |
| 0 | 2 | 21 |
| 0 | 252 | 311 |
G:=sub<GL(3,GF(313))| [312,0,0,0,312,0,0,0,312],[1,0,0,0,78,79,0,312,312],[288,0,0,0,2,252,0,21,311] >;
C2×C13⋊2C8 in GAP, Magma, Sage, TeX
C_2\times C_{13}\rtimes_2C_8 % in TeX
G:=Group("C2xC13:2C8"); // GroupNames label
G:=SmallGroup(208,9);
// by ID
G=gap.SmallGroup(208,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^13=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export