Copied to
clipboard

## G = D4.9D26order 416 = 25·13

### 4th non-split extension by D4 of D26 acting via D26/C26=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D4.9D26
 Chief series C1 — C13 — C26 — C52 — Dic26 — C2×Dic26 — D4.9D26
 Lower central C13 — C26 — C52 — D4.9D26
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4.9D26
G = < a,b,c,d | a4=b2=c26=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 312 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C13, M4(2), SD16, Q16, C2×Q8, C4○D4, C26, C26, C8.C22, Dic13, C52, C52, C2×C26, C2×C26, C132C8, Dic26, Dic26, C2×Dic13, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C52.4C4, D4.D13, C13⋊Q16, C2×Dic26, C13×C4○D4, D4.9D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8.C22, D26, C13⋊D4, C22×D13, C2×C13⋊D4, D4.9D26

Smallest permutation representation of D4.9D26
On 208 points
Generators in S208
```(1 42 97 24)(2 43 98 25)(3 44 99 26)(4 45 100 14)(5 46 101 15)(6 47 102 16)(7 48 103 17)(8 49 104 18)(9 50 92 19)(10 51 93 20)(11 52 94 21)(12 40 95 22)(13 41 96 23)(27 87 65 70)(28 88 53 71)(29 89 54 72)(30 90 55 73)(31 91 56 74)(32 79 57 75)(33 80 58 76)(34 81 59 77)(35 82 60 78)(36 83 61 66)(37 84 62 67)(38 85 63 68)(39 86 64 69)(105 164 118 177)(106 165 119 178)(107 166 120 179)(108 167 121 180)(109 168 122 181)(110 169 123 182)(111 170 124 157)(112 171 125 158)(113 172 126 159)(114 173 127 160)(115 174 128 161)(116 175 129 162)(117 176 130 163)(131 199 144 186)(132 200 145 187)(133 201 146 188)(134 202 147 189)(135 203 148 190)(136 204 149 191)(137 205 150 192)(138 206 151 193)(139 207 152 194)(140 208 153 195)(141 183 154 196)(142 184 155 197)(143 185 156 198)
(1 145)(2 133)(3 147)(4 135)(5 149)(6 137)(7 151)(8 139)(9 153)(10 141)(11 155)(12 143)(13 131)(14 203)(15 191)(16 205)(17 193)(18 207)(19 195)(20 183)(21 197)(22 185)(23 199)(24 187)(25 201)(26 189)(27 180)(28 168)(29 182)(30 170)(31 158)(32 172)(33 160)(34 174)(35 162)(36 176)(37 164)(38 178)(39 166)(40 198)(41 186)(42 200)(43 188)(44 202)(45 190)(46 204)(47 192)(48 206)(49 194)(50 208)(51 196)(52 184)(53 181)(54 169)(55 157)(56 171)(57 159)(58 173)(59 161)(60 175)(61 163)(62 177)(63 165)(64 179)(65 167)(66 130)(67 118)(68 106)(69 120)(70 108)(71 122)(72 110)(73 124)(74 112)(75 126)(76 114)(77 128)(78 116)(79 113)(80 127)(81 115)(82 129)(83 117)(84 105)(85 119)(86 107)(87 121)(88 109)(89 123)(90 111)(91 125)(92 140)(93 154)(94 142)(95 156)(96 144)(97 132)(98 146)(99 134)(100 148)(101 136)(102 150)(103 138)(104 152)
(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 78 97 82)(2 77 98 81)(3 76 99 80)(4 75 100 79)(5 74 101 91)(6 73 102 90)(7 72 103 89)(8 71 104 88)(9 70 92 87)(10 69 93 86)(11 68 94 85)(12 67 95 84)(13 66 96 83)(14 32 45 57)(15 31 46 56)(16 30 47 55)(17 29 48 54)(18 28 49 53)(19 27 50 65)(20 39 51 64)(21 38 52 63)(22 37 40 62)(23 36 41 61)(24 35 42 60)(25 34 43 59)(26 33 44 58)(105 185 118 198)(106 184 119 197)(107 183 120 196)(108 208 121 195)(109 207 122 194)(110 206 123 193)(111 205 124 192)(112 204 125 191)(113 203 126 190)(114 202 127 189)(115 201 128 188)(116 200 129 187)(117 199 130 186)(131 163 144 176)(132 162 145 175)(133 161 146 174)(134 160 147 173)(135 159 148 172)(136 158 149 171)(137 157 150 170)(138 182 151 169)(139 181 152 168)(140 180 153 167)(141 179 154 166)(142 178 155 165)(143 177 156 164)```

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

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

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

71 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 13A ··· 13F 26A ··· 26F 26G ··· 26X 52A ··· 52L 52M ··· 52AD order 1 2 2 2 4 4 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 2 4 2 2 4 52 52 52 52 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

71 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D13 D26 D26 D26 C13⋊D4 C13⋊D4 C8.C22 D4.9D26 kernel D4.9D26 C52.4C4 D4.D13 C13⋊Q16 C2×Dic26 C13×C4○D4 C52 C2×C26 C4○D4 C2×C4 D4 Q8 C4 C22 C13 C1 # reps 1 1 2 2 1 1 1 1 6 6 6 6 12 12 1 12

Matrix representation of D4.9D26 in GL4(𝔽313) generated by

 154 202 0 0 5 159 0 0 61 61 227 111 5 66 252 86
,
 79 301 150 139 0 103 86 146 3 100 91 12 114 160 173 40
,
 198 264 0 0 115 0 0 0 27 191 114 49 98 17 162 1
,
 6 11 0 0 82 307 0 0 26 132 4 14 192 222 200 309
`G:=sub<GL(4,GF(313))| [154,5,61,5,202,159,61,66,0,0,227,252,0,0,111,86],[79,0,3,114,301,103,100,160,150,86,91,173,139,146,12,40],[198,115,27,98,264,0,191,17,0,0,114,162,0,0,49,1],[6,82,26,192,11,307,132,222,0,0,4,200,0,0,14,309] >;`

D4.9D26 in GAP, Magma, Sage, TeX

`D_4._9D_{26}`
`% in TeX`

`G:=Group("D4.9D26");`
`// GroupNames label`

`G:=SmallGroup(416,172);`
`// by ID`

`G=gap.SmallGroup(416,172);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,218,188,579,159,69,13829]);`
`// Polycyclic`

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

׿
×
𝔽