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## G = D4.10D26order 416 = 25·13

### The non-split extension by D4 of D26 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4.10D26
 Chief series C1 — C13 — C26 — D26 — C4×D13 — Q8×D13 — D4.10D26
 Lower central C13 — C26 — D4.10D26
 Upper central C1 — C2 — C4○D4

Generators and relations for D4.10D26
G = < a,b,c,d | a4=b2=1, c26=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c25 >

Subgroups: 816 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, D4, Q8, Q8, C13, C2×Q8, C4○D4, C4○D4, D13, C26, C26, 2- 1+4, Dic13, C52, C52, D26, C2×C26, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, D4×C13, Q8×C13, C2×Dic26, D525C2, D42D13, Q8×D13, C13×C4○D4, D4.10D26
Quotients: C1, C2, C22, C23, C24, D13, 2- 1+4, D26, C22×D13, C23×D13, D4.10D26

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

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

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

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

77 conjugacy classes

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

77 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D13 D26 D26 D26 2- 1+4 D4.10D26 kernel D4.10D26 C2×Dic26 D52⋊5C2 D4⋊2D13 Q8×D13 C13×C4○D4 C4○D4 C2×C4 D4 Q8 C13 C1 # reps 1 3 3 6 2 1 6 18 18 6 1 12

Matrix representation of D4.10D26 in GL4(𝔽53) generated by

 28 0 1 18 0 28 52 29 40 30 25 0 16 16 0 25
,
 21 49 28 3 33 32 52 15 9 30 3 4 43 38 22 50
,
 31 46 20 39 18 37 43 10 2 45 11 7 1 19 12 27
,
 48 47 34 34 4 5 19 11 41 13 52 34 40 40 0 1
`G:=sub<GL(4,GF(53))| [28,0,40,16,0,28,30,16,1,52,25,0,18,29,0,25],[21,33,9,43,49,32,30,38,28,52,3,22,3,15,4,50],[31,18,2,1,46,37,45,19,20,43,11,12,39,10,7,27],[48,4,41,40,47,5,13,40,34,19,52,0,34,11,34,1] >;`

D4.10D26 in GAP, Magma, Sage, TeX

`D_4._{10}D_{26}`
`% in TeX`

`G:=Group("D4.10D26");`
`// GroupNames label`

`G:=SmallGroup(416,224);`
`// by ID`

`G=gap.SmallGroup(416,224);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,188,86,579,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^26=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^25>;`
`// generators/relations`

׿
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