Copied to
clipboard

## G = D26.12D4order 416 = 25·13

### 1st non-split extension by D26 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — D26.12D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — D26.12D4
 Lower central C13 — C2×C26 — D26.12D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D26.12D4
G = < a,b,c,d | a26=b2=c4=1, d2=a13, bab=a-1, ac=ca, ad=da, cbc-1=a13b, bd=db, dcd-1=a13c-1 >

Subgroups: 536 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22, C22 [×7], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C13, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, D13 [×2], C26 [×3], C26, C22.D4, Dic13 [×3], C52 [×2], D26 [×2], D26 [×2], C2×C26, C2×C26 [×3], C4×D13 [×2], C2×Dic13 [×3], C13⋊D4 [×2], C2×C52 [×2], C22×D13, C22×C26, C26.D4, C523C4, D26⋊C4, C23.D13, C13×C22⋊C4, C2×C4×D13, C2×C13⋊D4, D26.12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], D13, C22.D4, D26 [×3], C22×D13, D525C2, D4×D13, D42D13, D26.12D4

Smallest permutation representation of D26.12D4
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)
(1 169)(2 168)(3 167)(4 166)(5 165)(6 164)(7 163)(8 162)(9 161)(10 160)(11 159)(12 158)(13 157)(14 182)(15 181)(16 180)(17 179)(18 178)(19 177)(20 176)(21 175)(22 174)(23 173)(24 172)(25 171)(26 170)(27 192)(28 191)(29 190)(30 189)(31 188)(32 187)(33 186)(34 185)(35 184)(36 183)(37 208)(38 207)(39 206)(40 205)(41 204)(42 203)(43 202)(44 201)(45 200)(46 199)(47 198)(48 197)(49 196)(50 195)(51 194)(52 193)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)(61 105)(62 130)(63 129)(64 128)(65 127)(66 126)(67 125)(68 124)(69 123)(70 122)(71 121)(72 120)(73 119)(74 118)(75 117)(76 116)(77 115)(78 114)(79 131)(80 156)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 144)(93 143)(94 142)(95 141)(96 140)(97 139)(98 138)(99 137)(100 136)(101 135)(102 134)(103 133)(104 132)
(1 94 157 143)(2 95 158 144)(3 96 159 145)(4 97 160 146)(5 98 161 147)(6 99 162 148)(7 100 163 149)(8 101 164 150)(9 102 165 151)(10 103 166 152)(11 104 167 153)(12 79 168 154)(13 80 169 155)(14 81 170 156)(15 82 171 131)(16 83 172 132)(17 84 173 133)(18 85 174 134)(19 86 175 135)(20 87 176 136)(21 88 177 137)(22 89 178 138)(23 90 179 139)(24 91 180 140)(25 92 181 141)(26 93 182 142)(27 67 189 109)(28 68 190 110)(29 69 191 111)(30 70 192 112)(31 71 193 113)(32 72 194 114)(33 73 195 115)(34 74 196 116)(35 75 197 117)(36 76 198 118)(37 77 199 119)(38 78 200 120)(39 53 201 121)(40 54 202 122)(41 55 203 123)(42 56 204 124)(43 57 205 125)(44 58 206 126)(45 59 207 127)(46 60 208 128)(47 61 183 129)(48 62 184 130)(49 63 185 105)(50 64 186 106)(51 65 187 107)(52 66 188 108)
(1 124 14 111)(2 125 15 112)(3 126 16 113)(4 127 17 114)(5 128 18 115)(6 129 19 116)(7 130 20 117)(8 105 21 118)(9 106 22 119)(10 107 23 120)(11 108 24 121)(12 109 25 122)(13 110 26 123)(27 154 40 141)(28 155 41 142)(29 156 42 143)(30 131 43 144)(31 132 44 145)(32 133 45 146)(33 134 46 147)(34 135 47 148)(35 136 48 149)(36 137 49 150)(37 138 50 151)(38 139 51 152)(39 140 52 153)(53 167 66 180)(54 168 67 181)(55 169 68 182)(56 170 69 157)(57 171 70 158)(58 172 71 159)(59 173 72 160)(60 174 73 161)(61 175 74 162)(62 176 75 163)(63 177 76 164)(64 178 77 165)(65 179 78 166)(79 202 92 189)(80 203 93 190)(81 204 94 191)(82 205 95 192)(83 206 96 193)(84 207 97 194)(85 208 98 195)(86 183 99 196)(87 184 100 197)(88 185 101 198)(89 186 102 199)(90 187 103 200)(91 188 104 201)```

`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), (1,169)(2,168)(3,167)(4,166)(5,165)(6,164)(7,163)(8,162)(9,161)(10,160)(11,159)(12,158)(13,157)(14,182)(15,181)(16,180)(17,179)(18,178)(19,177)(20,176)(21,175)(22,174)(23,173)(24,172)(25,171)(26,170)(27,192)(28,191)(29,190)(30,189)(31,188)(32,187)(33,186)(34,185)(35,184)(36,183)(37,208)(38,207)(39,206)(40,205)(41,204)(42,203)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,196)(50,195)(51,194)(52,193)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106)(61,105)(62,130)(63,129)(64,128)(65,127)(66,126)(67,125)(68,124)(69,123)(70,122)(71,121)(72,120)(73,119)(74,118)(75,117)(76,116)(77,115)(78,114)(79,131)(80,156)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132), (1,94,157,143)(2,95,158,144)(3,96,159,145)(4,97,160,146)(5,98,161,147)(6,99,162,148)(7,100,163,149)(8,101,164,150)(9,102,165,151)(10,103,166,152)(11,104,167,153)(12,79,168,154)(13,80,169,155)(14,81,170,156)(15,82,171,131)(16,83,172,132)(17,84,173,133)(18,85,174,134)(19,86,175,135)(20,87,176,136)(21,88,177,137)(22,89,178,138)(23,90,179,139)(24,91,180,140)(25,92,181,141)(26,93,182,142)(27,67,189,109)(28,68,190,110)(29,69,191,111)(30,70,192,112)(31,71,193,113)(32,72,194,114)(33,73,195,115)(34,74,196,116)(35,75,197,117)(36,76,198,118)(37,77,199,119)(38,78,200,120)(39,53,201,121)(40,54,202,122)(41,55,203,123)(42,56,204,124)(43,57,205,125)(44,58,206,126)(45,59,207,127)(46,60,208,128)(47,61,183,129)(48,62,184,130)(49,63,185,105)(50,64,186,106)(51,65,187,107)(52,66,188,108), (1,124,14,111)(2,125,15,112)(3,126,16,113)(4,127,17,114)(5,128,18,115)(6,129,19,116)(7,130,20,117)(8,105,21,118)(9,106,22,119)(10,107,23,120)(11,108,24,121)(12,109,25,122)(13,110,26,123)(27,154,40,141)(28,155,41,142)(29,156,42,143)(30,131,43,144)(31,132,44,145)(32,133,45,146)(33,134,46,147)(34,135,47,148)(35,136,48,149)(36,137,49,150)(37,138,50,151)(38,139,51,152)(39,140,52,153)(53,167,66,180)(54,168,67,181)(55,169,68,182)(56,170,69,157)(57,171,70,158)(58,172,71,159)(59,173,72,160)(60,174,73,161)(61,175,74,162)(62,176,75,163)(63,177,76,164)(64,178,77,165)(65,179,78,166)(79,202,92,189)(80,203,93,190)(81,204,94,191)(82,205,95,192)(83,206,96,193)(84,207,97,194)(85,208,98,195)(86,183,99,196)(87,184,100,197)(88,185,101,198)(89,186,102,199)(90,187,103,200)(91,188,104,201)>;`

`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), (1,169)(2,168)(3,167)(4,166)(5,165)(6,164)(7,163)(8,162)(9,161)(10,160)(11,159)(12,158)(13,157)(14,182)(15,181)(16,180)(17,179)(18,178)(19,177)(20,176)(21,175)(22,174)(23,173)(24,172)(25,171)(26,170)(27,192)(28,191)(29,190)(30,189)(31,188)(32,187)(33,186)(34,185)(35,184)(36,183)(37,208)(38,207)(39,206)(40,205)(41,204)(42,203)(43,202)(44,201)(45,200)(46,199)(47,198)(48,197)(49,196)(50,195)(51,194)(52,193)(53,113)(54,112)(55,111)(56,110)(57,109)(58,108)(59,107)(60,106)(61,105)(62,130)(63,129)(64,128)(65,127)(66,126)(67,125)(68,124)(69,123)(70,122)(71,121)(72,120)(73,119)(74,118)(75,117)(76,116)(77,115)(78,114)(79,131)(80,156)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132), (1,94,157,143)(2,95,158,144)(3,96,159,145)(4,97,160,146)(5,98,161,147)(6,99,162,148)(7,100,163,149)(8,101,164,150)(9,102,165,151)(10,103,166,152)(11,104,167,153)(12,79,168,154)(13,80,169,155)(14,81,170,156)(15,82,171,131)(16,83,172,132)(17,84,173,133)(18,85,174,134)(19,86,175,135)(20,87,176,136)(21,88,177,137)(22,89,178,138)(23,90,179,139)(24,91,180,140)(25,92,181,141)(26,93,182,142)(27,67,189,109)(28,68,190,110)(29,69,191,111)(30,70,192,112)(31,71,193,113)(32,72,194,114)(33,73,195,115)(34,74,196,116)(35,75,197,117)(36,76,198,118)(37,77,199,119)(38,78,200,120)(39,53,201,121)(40,54,202,122)(41,55,203,123)(42,56,204,124)(43,57,205,125)(44,58,206,126)(45,59,207,127)(46,60,208,128)(47,61,183,129)(48,62,184,130)(49,63,185,105)(50,64,186,106)(51,65,187,107)(52,66,188,108), (1,124,14,111)(2,125,15,112)(3,126,16,113)(4,127,17,114)(5,128,18,115)(6,129,19,116)(7,130,20,117)(8,105,21,118)(9,106,22,119)(10,107,23,120)(11,108,24,121)(12,109,25,122)(13,110,26,123)(27,154,40,141)(28,155,41,142)(29,156,42,143)(30,131,43,144)(31,132,44,145)(32,133,45,146)(33,134,46,147)(34,135,47,148)(35,136,48,149)(36,137,49,150)(37,138,50,151)(38,139,51,152)(39,140,52,153)(53,167,66,180)(54,168,67,181)(55,169,68,182)(56,170,69,157)(57,171,70,158)(58,172,71,159)(59,173,72,160)(60,174,73,161)(61,175,74,162)(62,176,75,163)(63,177,76,164)(64,178,77,165)(65,179,78,166)(79,202,92,189)(80,203,93,190)(81,204,94,191)(82,205,95,192)(83,206,96,193)(84,207,97,194)(85,208,98,195)(86,183,99,196)(87,184,100,197)(88,185,101,198)(89,186,102,199)(90,187,103,200)(91,188,104,201) );`

`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)], [(1,169),(2,168),(3,167),(4,166),(5,165),(6,164),(7,163),(8,162),(9,161),(10,160),(11,159),(12,158),(13,157),(14,182),(15,181),(16,180),(17,179),(18,178),(19,177),(20,176),(21,175),(22,174),(23,173),(24,172),(25,171),(26,170),(27,192),(28,191),(29,190),(30,189),(31,188),(32,187),(33,186),(34,185),(35,184),(36,183),(37,208),(38,207),(39,206),(40,205),(41,204),(42,203),(43,202),(44,201),(45,200),(46,199),(47,198),(48,197),(49,196),(50,195),(51,194),(52,193),(53,113),(54,112),(55,111),(56,110),(57,109),(58,108),(59,107),(60,106),(61,105),(62,130),(63,129),(64,128),(65,127),(66,126),(67,125),(68,124),(69,123),(70,122),(71,121),(72,120),(73,119),(74,118),(75,117),(76,116),(77,115),(78,114),(79,131),(80,156),(81,155),(82,154),(83,153),(84,152),(85,151),(86,150),(87,149),(88,148),(89,147),(90,146),(91,145),(92,144),(93,143),(94,142),(95,141),(96,140),(97,139),(98,138),(99,137),(100,136),(101,135),(102,134),(103,133),(104,132)], [(1,94,157,143),(2,95,158,144),(3,96,159,145),(4,97,160,146),(5,98,161,147),(6,99,162,148),(7,100,163,149),(8,101,164,150),(9,102,165,151),(10,103,166,152),(11,104,167,153),(12,79,168,154),(13,80,169,155),(14,81,170,156),(15,82,171,131),(16,83,172,132),(17,84,173,133),(18,85,174,134),(19,86,175,135),(20,87,176,136),(21,88,177,137),(22,89,178,138),(23,90,179,139),(24,91,180,140),(25,92,181,141),(26,93,182,142),(27,67,189,109),(28,68,190,110),(29,69,191,111),(30,70,192,112),(31,71,193,113),(32,72,194,114),(33,73,195,115),(34,74,196,116),(35,75,197,117),(36,76,198,118),(37,77,199,119),(38,78,200,120),(39,53,201,121),(40,54,202,122),(41,55,203,123),(42,56,204,124),(43,57,205,125),(44,58,206,126),(45,59,207,127),(46,60,208,128),(47,61,183,129),(48,62,184,130),(49,63,185,105),(50,64,186,106),(51,65,187,107),(52,66,188,108)], [(1,124,14,111),(2,125,15,112),(3,126,16,113),(4,127,17,114),(5,128,18,115),(6,129,19,116),(7,130,20,117),(8,105,21,118),(9,106,22,119),(10,107,23,120),(11,108,24,121),(12,109,25,122),(13,110,26,123),(27,154,40,141),(28,155,41,142),(29,156,42,143),(30,131,43,144),(31,132,44,145),(32,133,45,146),(33,134,46,147),(34,135,47,148),(35,136,48,149),(36,137,49,150),(37,138,50,151),(38,139,51,152),(39,140,52,153),(53,167,66,180),(54,168,67,181),(55,169,68,182),(56,170,69,157),(57,171,70,158),(58,172,71,159),(59,173,72,160),(60,174,73,161),(61,175,74,162),(62,176,75,163),(63,177,76,164),(64,178,77,165),(65,179,78,166),(79,202,92,189),(80,203,93,190),(81,204,94,191),(82,205,95,192),(83,206,96,193),(84,207,97,194),(85,208,98,195),(86,183,99,196),(87,184,100,197),(88,185,101,198),(89,186,102,199),(90,187,103,200),(91,188,104,201)])`

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 13A ··· 13F 26A ··· 26R 26S ··· 26AD 52A ··· 52X order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 4 26 26 2 2 4 26 26 52 52 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 D13 D26 D26 D52⋊5C2 D4×D13 D4⋊2D13 kernel D26.12D4 C26.D4 C52⋊3C4 D26⋊C4 C23.D13 C13×C22⋊C4 C2×C4×D13 C2×C13⋊D4 D26 C26 C22⋊C4 C2×C4 C23 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 2 4 6 12 6 24 6 6

Matrix representation of D26.12D4 in GL4(𝔽53) generated by

 16 46 0 0 10 52 0 0 0 0 52 0 0 0 0 52
,
 26 31 0 0 9 27 0 0 0 0 1 0 0 0 27 52
,
 42 34 0 0 12 11 0 0 0 0 23 14 0 0 38 30
,
 30 0 0 0 0 30 0 0 0 0 23 0 0 0 38 30
`G:=sub<GL(4,GF(53))| [16,10,0,0,46,52,0,0,0,0,52,0,0,0,0,52],[26,9,0,0,31,27,0,0,0,0,1,27,0,0,0,52],[42,12,0,0,34,11,0,0,0,0,23,38,0,0,14,30],[30,0,0,0,0,30,0,0,0,0,23,38,0,0,0,30] >;`

D26.12D4 in GAP, Magma, Sage, TeX

`D_{26}._{12}D_4`
`% in TeX`

`G:=Group("D26.12D4");`
`// GroupNames label`

`G:=SmallGroup(416,104);`
`// by ID`

`G=gap.SmallGroup(416,104);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,55,218,188,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=a^13*c^-1>;`
`// generators/relations`

׿
×
𝔽