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## G = Q8.D22order 352 = 25·11

### 2nd non-split extension by Q8 of D22 acting via D22/D11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — Q8.D22
 Chief series C1 — C11 — C22 — C44 — C4×D11 — D4⋊2D11 — Q8.D22
 Lower central C11 — C22 — C44 — Q8.D22
 Upper central C1 — C2 — C4 — SD16

Generators and relations for Q8.D22
G = < a,b,c,d | a4=c22=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 402 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, Q8, C11, C2×C8, D8, SD16, SD16, Q16, C4○D4, D11, C22, C22, C4○D8, Dic11, Dic11, C44, C44, D22, D22, C2×C22, C11⋊C8, C88, Dic22, C4×D11, C4×D11, D44, D44, C2×Dic11, C11⋊D4, D4×C11, Q8×C11, C8×D11, C8⋊D11, D4⋊D11, C11⋊Q16, C11×SD16, D42D11, D44⋊C2, Q8.D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4○D8, D22, C22×D11, D4×D11, Q8.D22

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

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

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

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

49 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 11A ··· 11E 22A ··· 22E 22F ··· 22J 44A ··· 44E 44F ··· 44J 88A ··· 88J order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 size 1 1 4 22 44 2 4 11 11 44 2 2 22 22 2 ··· 2 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

49 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D11 C4○D8 D22 D22 D22 D4×D11 Q8.D22 kernel Q8.D22 C8×D11 C8⋊D11 D4⋊D11 C11⋊Q16 C11×SD16 D4⋊2D11 D44⋊C2 Dic11 D22 SD16 C11 C8 D4 Q8 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 5 4 5 5 5 5 10

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

 0 1 0 0 88 0 0 0 0 0 1 0 0 0 0 1
,
 0 34 0 0 34 0 0 0 0 0 1 0 0 0 0 1
,
 57 57 0 0 57 32 0 0 0 0 0 34 0 0 34 55
,
 20 69 0 0 69 69 0 0 0 0 34 34 0 0 21 55
`G:=sub<GL(4,GF(89))| [0,88,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,34,0,0,34,0,0,0,0,0,1,0,0,0,0,1],[57,57,0,0,57,32,0,0,0,0,0,34,0,0,34,55],[20,69,0,0,69,69,0,0,0,0,34,21,0,0,34,55] >;`

Q8.D22 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(352,111);`
`// by ID`

`G=gap.SmallGroup(352,111);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,362,116,86,297,159,69,11525]);`
`// Polycyclic`

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

׿
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