direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C26, C23.3C52, C104⋊14C22, C52.53C23, C8⋊4(C2×C26), (C2×C8)⋊6C26, (C2×C4).6C52, (C2×C104)⋊14C2, C4.10(C2×C52), (C2×C52).26C4, C52.68(C2×C4), C22.6(C2×C52), (C22×C26).8C4, C2.6(C22×C52), (C22×C4).6C26, (C22×C52).16C2, C26.47(C22×C4), C4.11(C22×C26), (C2×C52).127C22, (C2×C26).43(C2×C4), (C2×C4).33(C2×C26), SmallGroup(416,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C26
G = < a,b,c | a26=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C13, C2×C8, M4(2), C22×C4, C26, C26, C26, C2×M4(2), C52, C52, C2×C26, C2×C26, C2×C26, C104, C2×C52, C2×C52, C22×C26, C2×C104, C13×M4(2), C22×C52, M4(2)×C26
Quotients: C1, C2, C4, C22, C2×C4, C23, C13, M4(2), C22×C4, C26, C2×M4(2), C52, C2×C26, C2×C52, C22×C26, C13×M4(2), C22×C52, M4(2)×C26
►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 71 147 110 178 50 86 183)(2 72 148 111 179 51 87 184)(3 73 149 112 180 52 88 185)(4 74 150 113 181 27 89 186)(5 75 151 114 182 28 90 187)(6 76 152 115 157 29 91 188)(7 77 153 116 158 30 92 189)(8 78 154 117 159 31 93 190)(9 53 155 118 160 32 94 191)(10 54 156 119 161 33 95 192)(11 55 131 120 162 34 96 193)(12 56 132 121 163 35 97 194)(13 57 133 122 164 36 98 195)(14 58 134 123 165 37 99 196)(15 59 135 124 166 38 100 197)(16 60 136 125 167 39 101 198)(17 61 137 126 168 40 102 199)(18 62 138 127 169 41 103 200)(19 63 139 128 170 42 104 201)(20 64 140 129 171 43 79 202)(21 65 141 130 172 44 80 203)(22 66 142 105 173 45 81 204)(23 67 143 106 174 46 82 205)(24 68 144 107 175 47 83 206)(25 69 145 108 176 48 84 207)(26 70 146 109 177 49 85 208)
(1 165)(2 166)(3 167)(4 168)(5 169)(6 170)(7 171)(8 172)(9 173)(10 174)(11 175)(12 176)(13 177)(14 178)(15 179)(16 180)(17 181)(18 182)(19 157)(20 158)(21 159)(22 160)(23 161)(24 162)(25 163)(26 164)(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)(79 153)(80 154)(81 155)(82 156)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(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)(183 196)(184 197)(185 198)(186 199)(187 200)(188 201)(189 202)(190 203)(191 204)(192 205)(193 206)(194 207)(195 208)
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), (1,71,147,110,178,50,86,183)(2,72,148,111,179,51,87,184)(3,73,149,112,180,52,88,185)(4,74,150,113,181,27,89,186)(5,75,151,114,182,28,90,187)(6,76,152,115,157,29,91,188)(7,77,153,116,158,30,92,189)(8,78,154,117,159,31,93,190)(9,53,155,118,160,32,94,191)(10,54,156,119,161,33,95,192)(11,55,131,120,162,34,96,193)(12,56,132,121,163,35,97,194)(13,57,133,122,164,36,98,195)(14,58,134,123,165,37,99,196)(15,59,135,124,166,38,100,197)(16,60,136,125,167,39,101,198)(17,61,137,126,168,40,102,199)(18,62,138,127,169,41,103,200)(19,63,139,128,170,42,104,201)(20,64,140,129,171,43,79,202)(21,65,141,130,172,44,80,203)(22,66,142,105,173,45,81,204)(23,67,143,106,174,46,82,205)(24,68,144,107,175,47,83,206)(25,69,145,108,176,48,84,207)(26,70,146,109,177,49,85,208), (1,165)(2,166)(3,167)(4,168)(5,169)(6,170)(7,171)(8,172)(9,173)(10,174)(11,175)(12,176)(13,177)(14,178)(15,179)(16,180)(17,181)(18,182)(19,157)(20,158)(21,159)(22,160)(23,161)(24,162)(25,163)(26,164)(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)(79,153)(80,154)(81,155)(82,156)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(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)(183,196)(184,197)(185,198)(186,199)(187,200)(188,201)(189,202)(190,203)(191,204)(192,205)(193,206)(194,207)(195,208)>;
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), (1,71,147,110,178,50,86,183)(2,72,148,111,179,51,87,184)(3,73,149,112,180,52,88,185)(4,74,150,113,181,27,89,186)(5,75,151,114,182,28,90,187)(6,76,152,115,157,29,91,188)(7,77,153,116,158,30,92,189)(8,78,154,117,159,31,93,190)(9,53,155,118,160,32,94,191)(10,54,156,119,161,33,95,192)(11,55,131,120,162,34,96,193)(12,56,132,121,163,35,97,194)(13,57,133,122,164,36,98,195)(14,58,134,123,165,37,99,196)(15,59,135,124,166,38,100,197)(16,60,136,125,167,39,101,198)(17,61,137,126,168,40,102,199)(18,62,138,127,169,41,103,200)(19,63,139,128,170,42,104,201)(20,64,140,129,171,43,79,202)(21,65,141,130,172,44,80,203)(22,66,142,105,173,45,81,204)(23,67,143,106,174,46,82,205)(24,68,144,107,175,47,83,206)(25,69,145,108,176,48,84,207)(26,70,146,109,177,49,85,208), (1,165)(2,166)(3,167)(4,168)(5,169)(6,170)(7,171)(8,172)(9,173)(10,174)(11,175)(12,176)(13,177)(14,178)(15,179)(16,180)(17,181)(18,182)(19,157)(20,158)(21,159)(22,160)(23,161)(24,162)(25,163)(26,164)(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)(79,153)(80,154)(81,155)(82,156)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(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)(183,196)(184,197)(185,198)(186,199)(187,200)(188,201)(189,202)(190,203)(191,204)(192,205)(193,206)(194,207)(195,208) );
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)], [(1,71,147,110,178,50,86,183),(2,72,148,111,179,51,87,184),(3,73,149,112,180,52,88,185),(4,74,150,113,181,27,89,186),(5,75,151,114,182,28,90,187),(6,76,152,115,157,29,91,188),(7,77,153,116,158,30,92,189),(8,78,154,117,159,31,93,190),(9,53,155,118,160,32,94,191),(10,54,156,119,161,33,95,192),(11,55,131,120,162,34,96,193),(12,56,132,121,163,35,97,194),(13,57,133,122,164,36,98,195),(14,58,134,123,165,37,99,196),(15,59,135,124,166,38,100,197),(16,60,136,125,167,39,101,198),(17,61,137,126,168,40,102,199),(18,62,138,127,169,41,103,200),(19,63,139,128,170,42,104,201),(20,64,140,129,171,43,79,202),(21,65,141,130,172,44,80,203),(22,66,142,105,173,45,81,204),(23,67,143,106,174,46,82,205),(24,68,144,107,175,47,83,206),(25,69,145,108,176,48,84,207),(26,70,146,109,177,49,85,208)], [(1,165),(2,166),(3,167),(4,168),(5,169),(6,170),(7,171),(8,172),(9,173),(10,174),(11,175),(12,176),(13,177),(14,178),(15,179),(16,180),(17,181),(18,182),(19,157),(20,158),(21,159),(22,160),(23,161),(24,162),(25,163),(26,164),(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),(79,153),(80,154),(81,155),(82,156),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148),(101,149),(102,150),(103,151),(104,152),(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),(183,196),(184,197),(185,198),(186,199),(187,200),(188,201),(189,202),(190,203),(191,204),(192,205),(193,206),(194,207),(195,208)]])
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 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C13 | C26 | C26 | C26 | C52 | C52 | M4(2) | C13×M4(2) |
| kernel | M4(2)×C26 | C2×C104 | C13×M4(2) | C22×C52 | C2×C52 | C22×C26 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C26 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 24 | 48 | 12 | 72 | 24 | 4 | 48 |
Matrix representation of M4(2)×C26 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 48 | 0 |
| 0 | 0 | 48 |
| 25 | 0 | 0 |
| 0 | 139 | 311 |
| 0 | 283 | 174 |
| 1 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 174 | 1 |
G:=sub<GL(3,GF(313))| [312,0,0,0,48,0,0,0,48],[25,0,0,0,139,283,0,311,174],[1,0,0,0,312,174,0,0,1] >;
M4(2)×C26 in GAP, Magma, Sage, TeX
M_4(2)\times C_{26} % in TeX
G:=Group("M4(2)xC26"); // GroupNames label
G:=SmallGroup(416,191);
// by ID
G=gap.SmallGroup(416,191);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,624,2521,88]);
// Polycyclic
G:=Group<a,b,c|a^26=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations