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## G = Q8.10D26order 416 = 25·13

### 1st non-split extension by Q8 of D26 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8.10D26
G = < a,b,c,d | a4=1, b2=c26=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c25 >

Subgroups: 864 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2 [×5], C4 [×6], C4 [×4], C22, C22 [×4], C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], C13, C2×Q8, C2×Q8 [×4], C4○D4 [×10], D13 [×4], C26, C26, 2- 1+4, Dic13 [×4], C52 [×6], D26 [×4], C2×C26, Dic26 [×6], C4×D13 [×12], D52 [×6], C13⋊D4 [×4], C2×C52 [×3], Q8×C13 [×4], D525C2 [×6], Q8×D13 [×4], D52⋊C2 [×4], Q8×C26, Q8.10D26
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, D13, 2- 1+4, D26 [×7], C22×D13 [×7], C23×D13, Q8.10D26

Smallest permutation representation of Q8.10D26
On 208 points
Generators in S208
```(1 14 27 40)(2 15 28 41)(3 16 29 42)(4 17 30 43)(5 18 31 44)(6 19 32 45)(7 20 33 46)(8 21 34 47)(9 22 35 48)(10 23 36 49)(11 24 37 50)(12 25 38 51)(13 26 39 52)(53 66 79 92)(54 67 80 93)(55 68 81 94)(56 69 82 95)(57 70 83 96)(58 71 84 97)(59 72 85 98)(60 73 86 99)(61 74 87 100)(62 75 88 101)(63 76 89 102)(64 77 90 103)(65 78 91 104)(105 144 131 118)(106 145 132 119)(107 146 133 120)(108 147 134 121)(109 148 135 122)(110 149 136 123)(111 150 137 124)(112 151 138 125)(113 152 139 126)(114 153 140 127)(115 154 141 128)(116 155 142 129)(117 156 143 130)(157 196 183 170)(158 197 184 171)(159 198 185 172)(160 199 186 173)(161 200 187 174)(162 201 188 175)(163 202 189 176)(164 203 190 177)(165 204 191 178)(166 205 192 179)(167 206 193 180)(168 207 194 181)(169 208 195 182)
(1 76 27 102)(2 103 28 77)(3 78 29 104)(4 53 30 79)(5 80 31 54)(6 55 32 81)(7 82 33 56)(8 57 34 83)(9 84 35 58)(10 59 36 85)(11 86 37 60)(12 61 38 87)(13 88 39 62)(14 63 40 89)(15 90 41 64)(16 65 42 91)(17 92 43 66)(18 67 44 93)(19 94 45 68)(20 69 46 95)(21 96 47 70)(22 71 48 97)(23 98 49 72)(24 73 50 99)(25 100 51 74)(26 75 52 101)(105 201 131 175)(106 176 132 202)(107 203 133 177)(108 178 134 204)(109 205 135 179)(110 180 136 206)(111 207 137 181)(112 182 138 208)(113 157 139 183)(114 184 140 158)(115 159 141 185)(116 186 142 160)(117 161 143 187)(118 188 144 162)(119 163 145 189)(120 190 146 164)(121 165 147 191)(122 192 148 166)(123 167 149 193)(124 194 150 168)(125 169 151 195)(126 196 152 170)(127 171 153 197)(128 198 154 172)(129 173 155 199)(130 200 156 174)
(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)
(1 109 27 135)(2 134 28 108)(3 107 29 133)(4 132 30 106)(5 105 31 131)(6 130 32 156)(7 155 33 129)(8 128 34 154)(9 153 35 127)(10 126 36 152)(11 151 37 125)(12 124 38 150)(13 149 39 123)(14 122 40 148)(15 147 41 121)(16 120 42 146)(17 145 43 119)(18 118 44 144)(19 143 45 117)(20 116 46 142)(21 141 47 115)(22 114 48 140)(23 139 49 113)(24 112 50 138)(25 137 51 111)(26 110 52 136)(53 176 79 202)(54 201 80 175)(55 174 81 200)(56 199 82 173)(57 172 83 198)(58 197 84 171)(59 170 85 196)(60 195 86 169)(61 168 87 194)(62 193 88 167)(63 166 89 192)(64 191 90 165)(65 164 91 190)(66 189 92 163)(67 162 93 188)(68 187 94 161)(69 160 95 186)(70 185 96 159)(71 158 97 184)(72 183 98 157)(73 208 99 182)(74 181 100 207)(75 206 101 180)(76 179 102 205)(77 204 103 178)(78 177 104 203)```

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

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

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

77 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4F 4G 4H 4I 4J 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 2 2 2 4 ··· 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 2 26 26 26 26 2 ··· 2 26 26 26 26 2 ··· 2 2 ··· 2 4 ··· 4

77 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 D13 D26 D26 2- 1+4 Q8.10D26 kernel Q8.10D26 D52⋊5C2 Q8×D13 D52⋊C2 Q8×C26 C2×Q8 C2×C4 Q8 C13 C1 # reps 1 6 4 4 1 6 18 24 1 12

Matrix representation of Q8.10D26 in GL4(𝔽53) generated by

 1 12 44 0 44 52 14 39 0 0 0 52 0 0 1 0
,
 8 45 0 47 28 45 33 20 0 0 17 9 0 0 9 36
,
 42 27 25 32 46 11 5 48 0 0 0 24 0 0 29 0
,
 5 7 42 34 0 0 29 0 0 42 0 0 46 11 5 48
`G:=sub<GL(4,GF(53))| [1,44,0,0,12,52,0,0,44,14,0,1,0,39,52,0],[8,28,0,0,45,45,0,0,0,33,17,9,47,20,9,36],[42,46,0,0,27,11,0,0,25,5,0,29,32,48,24,0],[5,0,0,46,7,0,42,11,42,29,0,5,34,0,0,48] >;`

Q8.10D26 in GAP, Magma, Sage, TeX

`Q_8._{10}D_{26}`
`% in TeX`

`G:=Group("Q8.10D26");`
`// GroupNames label`

`G:=SmallGroup(416,221);`
`// by ID`

`G=gap.SmallGroup(416,221);`
`# by ID`

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

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

׿
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