metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C26.5M4(2), Dic13.23D4, (C2×C26)⋊1C8, C22⋊(C13⋊C8), C26.9(C2×C8), C13⋊2(C22⋊C8), C23.2(C13⋊C4), (C22×C26).3C4, C2.3(D13.D4), C26.10(C22⋊C4), (C2×Dic13).12C4, C2.3(C13⋊M4(2)), (C22×Dic13).7C2, (C2×Dic13).53C22, (C2×C13⋊C8)⋊2C2, C2.5(C2×C13⋊C8), (C2×C26).11(C2×C4), C22.14(C2×C13⋊C4), SmallGroup(416,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C26.M4(2)
G = < a,b,c | a26=b8=c2=1, bab-1=a5, ac=ca, cbc=a13b5 >
2C2
2C2
2C22
2C22
13C4
13C4
26C4
2C26
2C26
13C2×C4
13C2×C4
26C8
26C8
26C2×C4
26C2×C4
13C2×C8
13C22×C4
13C2×C8
13C22⋊C8
Smallest permutation representation of C26.M4(2)
►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)
(1 164 42 137 116 88 194 55)(2 159 41 142 117 83 193 60)(3 180 40 147 118 104 192 65)(4 175 39 152 119 99 191 70)(5 170 38 131 120 94 190 75)(6 165 37 136 121 89 189 54)(7 160 36 141 122 84 188 59)(8 181 35 146 123 79 187 64)(9 176 34 151 124 100 186 69)(10 171 33 156 125 95 185 74)(11 166 32 135 126 90 184 53)(12 161 31 140 127 85 183 58)(13 182 30 145 128 80 208 63)(14 177 29 150 129 101 207 68)(15 172 28 155 130 96 206 73)(16 167 27 134 105 91 205 78)(17 162 52 139 106 86 204 57)(18 157 51 144 107 81 203 62)(19 178 50 149 108 102 202 67)(20 173 49 154 109 97 201 72)(21 168 48 133 110 92 200 77)(22 163 47 138 111 87 199 56)(23 158 46 143 112 82 198 61)(24 179 45 148 113 103 197 66)(25 174 44 153 114 98 196 71)(26 169 43 132 115 93 195 76)
(53 148)(54 149)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 156)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 168)(80 169)(81 170)(82 171)(83 172)(84 173)(85 174)(86 175)(87 176)(88 177)(89 178)(90 179)(91 180)(92 181)(93 182)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)
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), (1,164,42,137,116,88,194,55)(2,159,41,142,117,83,193,60)(3,180,40,147,118,104,192,65)(4,175,39,152,119,99,191,70)(5,170,38,131,120,94,190,75)(6,165,37,136,121,89,189,54)(7,160,36,141,122,84,188,59)(8,181,35,146,123,79,187,64)(9,176,34,151,124,100,186,69)(10,171,33,156,125,95,185,74)(11,166,32,135,126,90,184,53)(12,161,31,140,127,85,183,58)(13,182,30,145,128,80,208,63)(14,177,29,150,129,101,207,68)(15,172,28,155,130,96,206,73)(16,167,27,134,105,91,205,78)(17,162,52,139,106,86,204,57)(18,157,51,144,107,81,203,62)(19,178,50,149,108,102,202,67)(20,173,49,154,109,97,201,72)(21,168,48,133,110,92,200,77)(22,163,47,138,111,87,199,56)(23,158,46,143,112,82,198,61)(24,179,45,148,113,103,197,66)(25,174,44,153,114,98,196,71)(26,169,43,132,115,93,195,76), (53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,168)(80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)>;
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), (1,164,42,137,116,88,194,55)(2,159,41,142,117,83,193,60)(3,180,40,147,118,104,192,65)(4,175,39,152,119,99,191,70)(5,170,38,131,120,94,190,75)(6,165,37,136,121,89,189,54)(7,160,36,141,122,84,188,59)(8,181,35,146,123,79,187,64)(9,176,34,151,124,100,186,69)(10,171,33,156,125,95,185,74)(11,166,32,135,126,90,184,53)(12,161,31,140,127,85,183,58)(13,182,30,145,128,80,208,63)(14,177,29,150,129,101,207,68)(15,172,28,155,130,96,206,73)(16,167,27,134,105,91,205,78)(17,162,52,139,106,86,204,57)(18,157,51,144,107,81,203,62)(19,178,50,149,108,102,202,67)(20,173,49,154,109,97,201,72)(21,168,48,133,110,92,200,77)(22,163,47,138,111,87,199,56)(23,158,46,143,112,82,198,61)(24,179,45,148,113,103,197,66)(25,174,44,153,114,98,196,71)(26,169,43,132,115,93,195,76), (53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,168)(80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167) );
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)], [(1,164,42,137,116,88,194,55),(2,159,41,142,117,83,193,60),(3,180,40,147,118,104,192,65),(4,175,39,152,119,99,191,70),(5,170,38,131,120,94,190,75),(6,165,37,136,121,89,189,54),(7,160,36,141,122,84,188,59),(8,181,35,146,123,79,187,64),(9,176,34,151,124,100,186,69),(10,171,33,156,125,95,185,74),(11,166,32,135,126,90,184,53),(12,161,31,140,127,85,183,58),(13,182,30,145,128,80,208,63),(14,177,29,150,129,101,207,68),(15,172,28,155,130,96,206,73),(16,167,27,134,105,91,205,78),(17,162,52,139,106,86,204,57),(18,157,51,144,107,81,203,62),(19,178,50,149,108,102,202,67),(20,173,49,154,109,97,201,72),(21,168,48,133,110,92,200,77),(22,163,47,138,111,87,199,56),(23,158,46,143,112,82,198,61),(24,179,45,148,113,103,197,66),(25,174,44,153,114,98,196,71),(26,169,43,132,115,93,195,76)], [(53,148),(54,149),(55,150),(56,151),(57,152),(58,153),(59,154),(60,155),(61,156),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,168),(80,169),(81,170),(82,171),(83,172),(84,173),(85,174),(86,175),(87,176),(88,177),(89,178),(90,179),(91,180),(92,181),(93,182),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 13A | 13B | 13C | 26A | ··· | 26U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | C13⋊C4 | C13⋊C8 | C2×C13⋊C4 | C13⋊M4(2) | D13.D4 |
| kernel | C26.M4(2) | C2×C13⋊C8 | C22×Dic13 | C2×Dic13 | C22×C26 | C2×C26 | Dic13 | C26 | C23 | C22 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 3 | 6 | 3 | 6 | 6 |
Matrix representation of C26.M4(2) ►in GL6(𝔽313)
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 266 | 266 | 0 | 0 |
| 0 | 0 | 47 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 288 | 162 |
| 0 | 0 | 0 | 0 | 151 | 286 |
| 288 | 4 | 0 | 0 | 0 | 0 |
| 157 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 77 | 77 | 0 | 0 |
| 0 | 0 | 49 | 236 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 169 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 312 | 0 |
| 0 | 0 | 0 | 0 | 0 | 312 |
G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,266,47,0,0,0,0,266,27,0,0,0,0,0,0,288,151,0,0,0,0,162,286],[288,157,0,0,0,0,4,25,0,0,0,0,0,0,0,0,77,49,0,0,0,0,77,236,0,0,1,0,0,0,0,0,0,1,0,0],[1,169,0,0,0,0,0,312,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,312,0,0,0,0,0,0,312] >;
C26.M4(2) in GAP, Magma, Sage, TeX
C_{26}.M_4(2) % in TeX
G:=Group("C26.M4(2)"); // GroupNames label
G:=SmallGroup(416,87);
// by ID
G=gap.SmallGroup(416,87);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^26=b^8=c^2=1,b*a*b^-1=a^5,a*c=c*a,c*b*c=a^13*b^5>;
// generators/relations
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