direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C13×C4⋊C4, C4⋊C52, C52⋊5C4, C26.3Q8, C26.13D4, C2.(Q8×C13), C2.2(C2×C52), (C2×C52).2C2, (C2×C4).1C26, C2.2(D4×C13), C26.18(C2×C4), C22.3(C2×C26), (C2×C26).14C22, SmallGroup(208,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4⋊C4
G = < a,b,c | a13=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×C4⋊C4
►Regular action 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)
(1 81 197 187)(2 82 198 188)(3 83 199 189)(4 84 200 190)(5 85 201 191)(6 86 202 192)(7 87 203 193)(8 88 204 194)(9 89 205 195)(10 90 206 183)(11 91 207 184)(12 79 208 185)(13 80 196 186)(14 143 182 101)(15 131 170 102)(16 132 171 103)(17 133 172 104)(18 134 173 92)(19 135 174 93)(20 136 175 94)(21 137 176 95)(22 138 177 96)(23 139 178 97)(24 140 179 98)(25 141 180 99)(26 142 181 100)(27 45 127 105)(28 46 128 106)(29 47 129 107)(30 48 130 108)(31 49 118 109)(32 50 119 110)(33 51 120 111)(34 52 121 112)(35 40 122 113)(36 41 123 114)(37 42 124 115)(38 43 125 116)(39 44 126 117)(53 152 71 160)(54 153 72 161)(55 154 73 162)(56 155 74 163)(57 156 75 164)(58 144 76 165)(59 145 77 166)(60 146 78 167)(61 147 66 168)(62 148 67 169)(63 149 68 157)(64 150 69 158)(65 151 70 159)
(1 56 132 38)(2 57 133 39)(3 58 134 27)(4 59 135 28)(5 60 136 29)(6 61 137 30)(7 62 138 31)(8 63 139 32)(9 64 140 33)(10 65 141 34)(11 53 142 35)(12 54 143 36)(13 55 131 37)(14 41 185 153)(15 42 186 154)(16 43 187 155)(17 44 188 156)(18 45 189 144)(19 46 190 145)(20 47 191 146)(21 48 192 147)(22 49 193 148)(23 50 194 149)(24 51 195 150)(25 52 183 151)(26 40 184 152)(66 95 130 202)(67 96 118 203)(68 97 119 204)(69 98 120 205)(70 99 121 206)(71 100 122 207)(72 101 123 208)(73 102 124 196)(74 103 125 197)(75 104 126 198)(76 92 127 199)(77 93 128 200)(78 94 129 201)(79 161 182 114)(80 162 170 115)(81 163 171 116)(82 164 172 117)(83 165 173 105)(84 166 174 106)(85 167 175 107)(86 168 176 108)(87 169 177 109)(88 157 178 110)(89 158 179 111)(90 159 180 112)(91 160 181 113)
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), (1,81,197,187)(2,82,198,188)(3,83,199,189)(4,84,200,190)(5,85,201,191)(6,86,202,192)(7,87,203,193)(8,88,204,194)(9,89,205,195)(10,90,206,183)(11,91,207,184)(12,79,208,185)(13,80,196,186)(14,143,182,101)(15,131,170,102)(16,132,171,103)(17,133,172,104)(18,134,173,92)(19,135,174,93)(20,136,175,94)(21,137,176,95)(22,138,177,96)(23,139,178,97)(24,140,179,98)(25,141,180,99)(26,142,181,100)(27,45,127,105)(28,46,128,106)(29,47,129,107)(30,48,130,108)(31,49,118,109)(32,50,119,110)(33,51,120,111)(34,52,121,112)(35,40,122,113)(36,41,123,114)(37,42,124,115)(38,43,125,116)(39,44,126,117)(53,152,71,160)(54,153,72,161)(55,154,73,162)(56,155,74,163)(57,156,75,164)(58,144,76,165)(59,145,77,166)(60,146,78,167)(61,147,66,168)(62,148,67,169)(63,149,68,157)(64,150,69,158)(65,151,70,159), (1,56,132,38)(2,57,133,39)(3,58,134,27)(4,59,135,28)(5,60,136,29)(6,61,137,30)(7,62,138,31)(8,63,139,32)(9,64,140,33)(10,65,141,34)(11,53,142,35)(12,54,143,36)(13,55,131,37)(14,41,185,153)(15,42,186,154)(16,43,187,155)(17,44,188,156)(18,45,189,144)(19,46,190,145)(20,47,191,146)(21,48,192,147)(22,49,193,148)(23,50,194,149)(24,51,195,150)(25,52,183,151)(26,40,184,152)(66,95,130,202)(67,96,118,203)(68,97,119,204)(69,98,120,205)(70,99,121,206)(71,100,122,207)(72,101,123,208)(73,102,124,196)(74,103,125,197)(75,104,126,198)(76,92,127,199)(77,93,128,200)(78,94,129,201)(79,161,182,114)(80,162,170,115)(81,163,171,116)(82,164,172,117)(83,165,173,105)(84,166,174,106)(85,167,175,107)(86,168,176,108)(87,169,177,109)(88,157,178,110)(89,158,179,111)(90,159,180,112)(91,160,181,113)>;
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), (1,81,197,187)(2,82,198,188)(3,83,199,189)(4,84,200,190)(5,85,201,191)(6,86,202,192)(7,87,203,193)(8,88,204,194)(9,89,205,195)(10,90,206,183)(11,91,207,184)(12,79,208,185)(13,80,196,186)(14,143,182,101)(15,131,170,102)(16,132,171,103)(17,133,172,104)(18,134,173,92)(19,135,174,93)(20,136,175,94)(21,137,176,95)(22,138,177,96)(23,139,178,97)(24,140,179,98)(25,141,180,99)(26,142,181,100)(27,45,127,105)(28,46,128,106)(29,47,129,107)(30,48,130,108)(31,49,118,109)(32,50,119,110)(33,51,120,111)(34,52,121,112)(35,40,122,113)(36,41,123,114)(37,42,124,115)(38,43,125,116)(39,44,126,117)(53,152,71,160)(54,153,72,161)(55,154,73,162)(56,155,74,163)(57,156,75,164)(58,144,76,165)(59,145,77,166)(60,146,78,167)(61,147,66,168)(62,148,67,169)(63,149,68,157)(64,150,69,158)(65,151,70,159), (1,56,132,38)(2,57,133,39)(3,58,134,27)(4,59,135,28)(5,60,136,29)(6,61,137,30)(7,62,138,31)(8,63,139,32)(9,64,140,33)(10,65,141,34)(11,53,142,35)(12,54,143,36)(13,55,131,37)(14,41,185,153)(15,42,186,154)(16,43,187,155)(17,44,188,156)(18,45,189,144)(19,46,190,145)(20,47,191,146)(21,48,192,147)(22,49,193,148)(23,50,194,149)(24,51,195,150)(25,52,183,151)(26,40,184,152)(66,95,130,202)(67,96,118,203)(68,97,119,204)(69,98,120,205)(70,99,121,206)(71,100,122,207)(72,101,123,208)(73,102,124,196)(74,103,125,197)(75,104,126,198)(76,92,127,199)(77,93,128,200)(78,94,129,201)(79,161,182,114)(80,162,170,115)(81,163,171,116)(82,164,172,117)(83,165,173,105)(84,166,174,106)(85,167,175,107)(86,168,176,108)(87,169,177,109)(88,157,178,110)(89,158,179,111)(90,159,180,112)(91,160,181,113) );
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)], [(1,81,197,187),(2,82,198,188),(3,83,199,189),(4,84,200,190),(5,85,201,191),(6,86,202,192),(7,87,203,193),(8,88,204,194),(9,89,205,195),(10,90,206,183),(11,91,207,184),(12,79,208,185),(13,80,196,186),(14,143,182,101),(15,131,170,102),(16,132,171,103),(17,133,172,104),(18,134,173,92),(19,135,174,93),(20,136,175,94),(21,137,176,95),(22,138,177,96),(23,139,178,97),(24,140,179,98),(25,141,180,99),(26,142,181,100),(27,45,127,105),(28,46,128,106),(29,47,129,107),(30,48,130,108),(31,49,118,109),(32,50,119,110),(33,51,120,111),(34,52,121,112),(35,40,122,113),(36,41,123,114),(37,42,124,115),(38,43,125,116),(39,44,126,117),(53,152,71,160),(54,153,72,161),(55,154,73,162),(56,155,74,163),(57,156,75,164),(58,144,76,165),(59,145,77,166),(60,146,78,167),(61,147,66,168),(62,148,67,169),(63,149,68,157),(64,150,69,158),(65,151,70,159)], [(1,56,132,38),(2,57,133,39),(3,58,134,27),(4,59,135,28),(5,60,136,29),(6,61,137,30),(7,62,138,31),(8,63,139,32),(9,64,140,33),(10,65,141,34),(11,53,142,35),(12,54,143,36),(13,55,131,37),(14,41,185,153),(15,42,186,154),(16,43,187,155),(17,44,188,156),(18,45,189,144),(19,46,190,145),(20,47,191,146),(21,48,192,147),(22,49,193,148),(23,50,194,149),(24,51,195,150),(25,52,183,151),(26,40,184,152),(66,95,130,202),(67,96,118,203),(68,97,119,204),(69,98,120,205),(70,99,121,206),(71,100,122,207),(72,101,123,208),(73,102,124,196),(74,103,125,197),(75,104,126,198),(76,92,127,199),(77,93,128,200),(78,94,129,201),(79,161,182,114),(80,162,170,115),(81,163,171,116),(82,164,172,117),(83,165,173,105),(84,166,174,106),(85,167,175,107),(86,168,176,108),(87,169,177,109),(88,157,178,110),(89,158,179,111),(90,159,180,112),(91,160,181,113)]])
C13×C4⋊C4 is a maximal subgroup of
C26.D8 C52.Q8 D52⋊6C4 C26.Q16 Dic13⋊3Q8 C52⋊Q8 Dic13.Q8 C4.Dic26 C4⋊C4⋊7D13 D52⋊8C4 D26.13D4 C4⋊2D52 D26⋊Q8 D26⋊2Q8 C4⋊C4⋊D13 D4×C52 Q8×C52
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C13 | C26 | C52 | D4 | Q8 | D4×C13 | Q8×C13 |
| kernel | C13×C4⋊C4 | C2×C52 | C52 | C4⋊C4 | C2×C4 | C4 | C26 | C26 | C2 | C2 |
| # reps | 1 | 3 | 4 | 12 | 36 | 48 | 1 | 1 | 12 | 12 |
Matrix representation of C13×C4⋊C4 ►in GL3(𝔽53) generated by
| 1 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 52 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 52 | 0 |
| 23 | 0 | 0 |
| 0 | 31 | 10 |
| 0 | 10 | 22 |
G:=sub<GL(3,GF(53))| [1,0,0,0,46,0,0,0,46],[52,0,0,0,0,52,0,1,0],[23,0,0,0,31,10,0,10,22] >;
C13×C4⋊C4 in GAP, Magma, Sage, TeX
C_{13}\times C_4\rtimes C_4 % in TeX
G:=Group("C13xC4:C4"); // GroupNames label
G:=SmallGroup(208,22);
// by ID
G=gap.SmallGroup(208,22);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-2,520,541,266]);
// Polycyclic
G:=Group<a,b,c|a^13=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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