direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C22⋊C8, C22⋊C104, C52.65D4, C23.2C52, C26.13M4(2), (C2×C26)⋊3C8, (C2×C8)⋊1C26, (C2×C104)⋊3C2, (C2×C4).3C52, (C2×C52).17C4, C26.20(C2×C8), C2.1(C2×C104), C4.16(D4×C13), (C22×C26).7C4, (C22×C52).3C2, (C22×C4).2C26, C22.9(C2×C52), C2.2(C13×M4(2)), C26.31(C22⋊C4), (C2×C52).135C22, (C2×C26).58(C2×C4), (C2×C4).31(C2×C26), C2.2(C13×C22⋊C4), SmallGroup(416,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C22⋊C8
G = < a,b,c,d | a13=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C13×C22⋊C8
►On 208 points
Generators in S208
(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)
(14 127)(15 128)(16 129)(17 130)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(79 191)(80 192)(81 193)(82 194)(83 195)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 131)(101 132)(102 133)(103 134)(104 135)(144 182)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 181)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 53)(14 127)(15 128)(16 129)(17 130)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 117)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 158)(41 159)(42 160)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 169)(52 157)(66 199)(67 200)(68 201)(69 202)(70 203)(71 204)(72 205)(73 206)(74 207)(75 208)(76 196)(77 197)(78 198)(79 191)(80 192)(81 193)(82 194)(83 195)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 131)(101 132)(102 133)(103 134)(104 135)(144 182)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 181)
(1 139 33 124 78 177 166 86)(2 140 34 125 66 178 167 87)(3 141 35 126 67 179 168 88)(4 142 36 127 68 180 169 89)(5 143 37 128 69 181 157 90)(6 131 38 129 70 182 158 91)(7 132 39 130 71 170 159 79)(8 133 27 118 72 171 160 80)(9 134 28 119 73 172 161 81)(10 135 29 120 74 173 162 82)(11 136 30 121 75 174 163 83)(12 137 31 122 76 175 164 84)(13 138 32 123 77 176 165 85)(14 201 155 51 188 57 98 113)(15 202 156 52 189 58 99 114)(16 203 144 40 190 59 100 115)(17 204 145 41 191 60 101 116)(18 205 146 42 192 61 102 117)(19 206 147 43 193 62 103 105)(20 207 148 44 194 63 104 106)(21 208 149 45 195 64 92 107)(22 196 150 46 183 65 93 108)(23 197 151 47 184 53 94 109)(24 198 152 48 185 54 95 110)(25 199 153 49 186 55 96 111)(26 200 154 50 187 56 97 112)
G:=sub<Sym(208)| (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), (14,127)(15,128)(16,129)(17,130)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(79,191)(80,192)(81,193)(82,194)(83,195)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,131)(101,132)(102,133)(103,134)(104,135)(144,182)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,53)(14,127)(15,128)(16,129)(17,130)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,117)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,157)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,196)(77,197)(78,198)(79,191)(80,192)(81,193)(82,194)(83,195)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,131)(101,132)(102,133)(103,134)(104,135)(144,182)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181), (1,139,33,124,78,177,166,86)(2,140,34,125,66,178,167,87)(3,141,35,126,67,179,168,88)(4,142,36,127,68,180,169,89)(5,143,37,128,69,181,157,90)(6,131,38,129,70,182,158,91)(7,132,39,130,71,170,159,79)(8,133,27,118,72,171,160,80)(9,134,28,119,73,172,161,81)(10,135,29,120,74,173,162,82)(11,136,30,121,75,174,163,83)(12,137,31,122,76,175,164,84)(13,138,32,123,77,176,165,85)(14,201,155,51,188,57,98,113)(15,202,156,52,189,58,99,114)(16,203,144,40,190,59,100,115)(17,204,145,41,191,60,101,116)(18,205,146,42,192,61,102,117)(19,206,147,43,193,62,103,105)(20,207,148,44,194,63,104,106)(21,208,149,45,195,64,92,107)(22,196,150,46,183,65,93,108)(23,197,151,47,184,53,94,109)(24,198,152,48,185,54,95,110)(25,199,153,49,186,55,96,111)(26,200,154,50,187,56,97,112)>;
G:=Group( (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), (14,127)(15,128)(16,129)(17,130)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(79,191)(80,192)(81,193)(82,194)(83,195)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,131)(101,132)(102,133)(103,134)(104,135)(144,182)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,53)(14,127)(15,128)(16,129)(17,130)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,117)(28,105)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,158)(41,159)(42,160)(43,161)(44,162)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,157)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,196)(77,197)(78,198)(79,191)(80,192)(81,193)(82,194)(83,195)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,131)(101,132)(102,133)(103,134)(104,135)(144,182)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181), (1,139,33,124,78,177,166,86)(2,140,34,125,66,178,167,87)(3,141,35,126,67,179,168,88)(4,142,36,127,68,180,169,89)(5,143,37,128,69,181,157,90)(6,131,38,129,70,182,158,91)(7,132,39,130,71,170,159,79)(8,133,27,118,72,171,160,80)(9,134,28,119,73,172,161,81)(10,135,29,120,74,173,162,82)(11,136,30,121,75,174,163,83)(12,137,31,122,76,175,164,84)(13,138,32,123,77,176,165,85)(14,201,155,51,188,57,98,113)(15,202,156,52,189,58,99,114)(16,203,144,40,190,59,100,115)(17,204,145,41,191,60,101,116)(18,205,146,42,192,61,102,117)(19,206,147,43,193,62,103,105)(20,207,148,44,194,63,104,106)(21,208,149,45,195,64,92,107)(22,196,150,46,183,65,93,108)(23,197,151,47,184,53,94,109)(24,198,152,48,185,54,95,110)(25,199,153,49,186,55,96,111)(26,200,154,50,187,56,97,112) );
G=PermutationGroup([[(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)], [(14,127),(15,128),(16,129),(17,130),(18,118),(19,119),(20,120),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(79,191),(80,192),(81,193),(82,194),(83,195),(84,183),(85,184),(86,185),(87,186),(88,187),(89,188),(90,189),(91,190),(92,136),(93,137),(94,138),(95,139),(96,140),(97,141),(98,142),(99,143),(100,131),(101,132),(102,133),(103,134),(104,135),(144,182),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,177),(153,178),(154,179),(155,180),(156,181)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,53),(14,127),(15,128),(16,129),(17,130),(18,118),(19,119),(20,120),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,117),(28,105),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,158),(41,159),(42,160),(43,161),(44,162),(45,163),(46,164),(47,165),(48,166),(49,167),(50,168),(51,169),(52,157),(66,199),(67,200),(68,201),(69,202),(70,203),(71,204),(72,205),(73,206),(74,207),(75,208),(76,196),(77,197),(78,198),(79,191),(80,192),(81,193),(82,194),(83,195),(84,183),(85,184),(86,185),(87,186),(88,187),(89,188),(90,189),(91,190),(92,136),(93,137),(94,138),(95,139),(96,140),(97,141),(98,142),(99,143),(100,131),(101,132),(102,133),(103,134),(104,135),(144,182),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,177),(153,178),(154,179),(155,180),(156,181)], [(1,139,33,124,78,177,166,86),(2,140,34,125,66,178,167,87),(3,141,35,126,67,179,168,88),(4,142,36,127,68,180,169,89),(5,143,37,128,69,181,157,90),(6,131,38,129,70,182,158,91),(7,132,39,130,71,170,159,79),(8,133,27,118,72,171,160,80),(9,134,28,119,73,172,161,81),(10,135,29,120,74,173,162,82),(11,136,30,121,75,174,163,83),(12,137,31,122,76,175,164,84),(13,138,32,123,77,176,165,85),(14,201,155,51,188,57,98,113),(15,202,156,52,189,58,99,114),(16,203,144,40,190,59,100,115),(17,204,145,41,191,60,101,116),(18,205,146,42,192,61,102,117),(19,206,147,43,193,62,103,105),(20,207,148,44,194,63,104,106),(21,208,149,45,195,64,92,107),(22,196,150,46,183,65,93,108),(23,197,151,47,184,53,94,109),(24,198,152,48,185,54,95,110),(25,199,153,49,186,55,96,111),(26,200,154,50,187,56,97,112)]])
260 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26BH | 52A | ··· | 52AV | 52AW | ··· | 52BT | 104A | ··· | 104CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
260 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C13 | C26 | C26 | C52 | C52 | C104 | D4 | M4(2) | D4×C13 | C13×M4(2) |
| kernel | C13×C22⋊C8 | C2×C104 | C22×C52 | C2×C52 | C22×C26 | C2×C26 | C22⋊C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C52 | C26 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 12 | 24 | 12 | 24 | 24 | 96 | 2 | 2 | 24 | 24 |
Matrix representation of C13×C22⋊C8 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 48 | 0 |
| 0 | 0 | 48 |
| 312 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 125 | 312 |
| 1 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 312 |
| 308 | 0 | 0 |
| 0 | 125 | 311 |
| 0 | 0 | 188 |
G:=sub<GL(3,GF(313))| [1,0,0,0,48,0,0,0,48],[312,0,0,0,1,125,0,0,312],[1,0,0,0,312,0,0,0,312],[308,0,0,0,125,0,0,311,188] >;
C13×C22⋊C8 in GAP, Magma, Sage, TeX
C_{13}\times C_2^2\rtimes C_8 % in TeX
G:=Group("C13xC2^2:C8"); // GroupNames label
G:=SmallGroup(416,48);
// by ID
G=gap.SmallGroup(416,48);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,88]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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