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## G = Q8.10D22order 352 = 25·11

### 1st non-split extension by Q8 of D22 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — Q8.10D22
 Chief series C1 — C11 — C22 — D22 — C4×D11 — Q8×D11 — Q8.10D22
 Lower central C11 — C22 — Q8.10D22
 Upper central C1 — C2 — C2×Q8

Generators and relations for Q8.10D22
G = < a,b,c,d | a4=1, b2=c22=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c21 >

Subgroups: 746 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, Q8, C11, C2×Q8, C2×Q8, C4○D4, D11, C22, C22, 2- 1+4, Dic11, C44, D22, C2×C22, Dic22, C4×D11, D44, C11⋊D4, C2×C44, Q8×C11, D445C2, Q8×D11, D44⋊C2, Q8×C22, Q8.10D22
Quotients: C1, C2, C22, C23, C24, D11, 2- 1+4, D22, C22×D11, C23×D11, Q8.10D22

Smallest permutation representation of Q8.10D22
On 176 points
Generators in S176
```(1 47 23 69)(2 48 24 70)(3 49 25 71)(4 50 26 72)(5 51 27 73)(6 52 28 74)(7 53 29 75)(8 54 30 76)(9 55 31 77)(10 56 32 78)(11 57 33 79)(12 58 34 80)(13 59 35 81)(14 60 36 82)(15 61 37 83)(16 62 38 84)(17 63 39 85)(18 64 40 86)(19 65 41 87)(20 66 42 88)(21 67 43 45)(22 68 44 46)(89 157 111 135)(90 158 112 136)(91 159 113 137)(92 160 114 138)(93 161 115 139)(94 162 116 140)(95 163 117 141)(96 164 118 142)(97 165 119 143)(98 166 120 144)(99 167 121 145)(100 168 122 146)(101 169 123 147)(102 170 124 148)(103 171 125 149)(104 172 126 150)(105 173 127 151)(106 174 128 152)(107 175 129 153)(108 176 130 154)(109 133 131 155)(110 134 132 156)
(1 124 23 102)(2 103 24 125)(3 126 25 104)(4 105 26 127)(5 128 27 106)(6 107 28 129)(7 130 29 108)(8 109 30 131)(9 132 31 110)(10 111 32 89)(11 90 33 112)(12 113 34 91)(13 92 35 114)(14 115 36 93)(15 94 37 116)(16 117 38 95)(17 96 39 118)(18 119 40 97)(19 98 41 120)(20 121 42 99)(21 100 43 122)(22 123 44 101)(45 168 67 146)(46 147 68 169)(47 170 69 148)(48 149 70 171)(49 172 71 150)(50 151 72 173)(51 174 73 152)(52 153 74 175)(53 176 75 154)(54 155 76 133)(55 134 77 156)(56 157 78 135)(57 136 79 158)(58 159 80 137)(59 138 81 160)(60 161 82 139)(61 140 83 162)(62 163 84 141)(63 142 85 164)(64 165 86 143)(65 144 87 166)(66 167 88 145)
(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)
(1 147 23 169)(2 168 24 146)(3 145 25 167)(4 166 26 144)(5 143 27 165)(6 164 28 142)(7 141 29 163)(8 162 30 140)(9 139 31 161)(10 160 32 138)(11 137 33 159)(12 158 34 136)(13 135 35 157)(14 156 36 134)(15 133 37 155)(16 154 38 176)(17 175 39 153)(18 152 40 174)(19 173 41 151)(20 150 42 172)(21 171 43 149)(22 148 44 170)(45 125 67 103)(46 102 68 124)(47 123 69 101)(48 100 70 122)(49 121 71 99)(50 98 72 120)(51 119 73 97)(52 96 74 118)(53 117 75 95)(54 94 76 116)(55 115 77 93)(56 92 78 114)(57 113 79 91)(58 90 80 112)(59 111 81 89)(60 132 82 110)(61 109 83 131)(62 130 84 108)(63 107 85 129)(64 128 86 106)(65 105 87 127)(66 126 88 104)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4F 4G 4H 4I 4J 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 2 2 2 4 ··· 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 2 22 22 22 22 2 ··· 2 22 22 22 22 2 ··· 2 2 ··· 2 4 ··· 4

67 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 D11 D22 D22 2- 1+4 Q8.10D22 kernel Q8.10D22 D44⋊5C2 Q8×D11 D44⋊C2 Q8×C22 C2×Q8 C2×C4 Q8 C11 C1 # reps 1 6 4 4 1 5 15 20 1 10

Matrix representation of Q8.10D22 in GL4(𝔽89) generated by

 1 58 38 44 0 88 32 16 57 88 61 31 64 13 58 28
,
 0 3 34 51 0 49 17 53 72 69 83 86 34 82 3 46
,
 57 38 31 61 0 25 1 45 45 24 45 51 88 33 38 51
,
 0 3 48 79 0 0 55 21 47 69 12 24 21 82 83 77
`G:=sub<GL(4,GF(89))| [1,0,57,64,58,88,88,13,38,32,61,58,44,16,31,28],[0,0,72,34,3,49,69,82,34,17,83,3,51,53,86,46],[57,0,45,88,38,25,24,33,31,1,45,38,61,45,51,51],[0,0,47,21,3,0,69,82,48,55,12,83,79,21,24,77] >;`

Q8.10D22 in GAP, Magma, Sage, TeX

`Q_8._{10}D_{22}`
`% in TeX`

`G:=Group("Q8.10D22");`
`// GroupNames label`

`G:=SmallGroup(352,182);`
`// by ID`

`G=gap.SmallGroup(352,182);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,188,86,579,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=1,b^2=c^22=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^21>;`
`// generators/relations`

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