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## G = D22⋊2Q8order 352 = 25·11

### 2nd semidirect product of D22 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D222Q8
G = < a,b,c,d | a22=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >

Subgroups: 450 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, Q8, C23, C11, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, D11, C22, C22⋊Q8, Dic11, C44, C44, D22, D22, C2×C22, Dic22, C4×D11, C2×Dic11, C2×Dic11, C2×C44, C2×C44, C22×D11, C44⋊C4, D22⋊C4, C11×C4⋊C4, C2×Dic22, C2×C4×D11, D222Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D11, C22⋊Q8, D22, D44, C22×D11, C2×D44, D42D11, Q8×D11, D222Q8

Smallest permutation representation of D222Q8
On 176 points
Generators in S176
```(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 44)(19 43)(20 42)(21 41)(22 40)(45 168)(46 167)(47 166)(48 165)(49 164)(50 163)(51 162)(52 161)(53 160)(54 159)(55 158)(56 157)(57 156)(58 155)(59 176)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(111 141)(112 140)(113 139)(114 138)(115 137)(116 136)(117 135)(118 134)(119 133)(120 154)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)
(1 80 40 94)(2 79 41 93)(3 78 42 92)(4 77 43 91)(5 76 44 90)(6 75 23 89)(7 74 24 110)(8 73 25 109)(9 72 26 108)(10 71 27 107)(11 70 28 106)(12 69 29 105)(13 68 30 104)(14 67 31 103)(15 88 32 102)(16 87 33 101)(17 86 34 100)(18 85 35 99)(19 84 36 98)(20 83 37 97)(21 82 38 96)(22 81 39 95)(45 123 162 148)(46 122 163 147)(47 121 164 146)(48 120 165 145)(49 119 166 144)(50 118 167 143)(51 117 168 142)(52 116 169 141)(53 115 170 140)(54 114 171 139)(55 113 172 138)(56 112 173 137)(57 111 174 136)(58 132 175 135)(59 131 176 134)(60 130 155 133)(61 129 156 154)(62 128 157 153)(63 127 158 152)(64 126 159 151)(65 125 160 150)(66 124 161 149)
(1 139 40 114)(2 140 41 115)(3 141 42 116)(4 142 43 117)(5 143 44 118)(6 144 23 119)(7 145 24 120)(8 146 25 121)(9 147 26 122)(10 148 27 123)(11 149 28 124)(12 150 29 125)(13 151 30 126)(14 152 31 127)(15 153 32 128)(16 154 33 129)(17 133 34 130)(18 134 35 131)(19 135 36 132)(20 136 37 111)(21 137 38 112)(22 138 39 113)(45 71 162 107)(46 72 163 108)(47 73 164 109)(48 74 165 110)(49 75 166 89)(50 76 167 90)(51 77 168 91)(52 78 169 92)(53 79 170 93)(54 80 171 94)(55 81 172 95)(56 82 173 96)(57 83 174 97)(58 84 175 98)(59 85 176 99)(60 86 155 100)(61 87 156 101)(62 88 157 102)(63 67 158 103)(64 68 159 104)(65 69 160 105)(66 70 161 106)```

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 D4 Q8 C4○D4 D11 D22 D44 D4⋊2D11 Q8×D11 kernel D22⋊2Q8 C44⋊C4 D22⋊C4 C11×C4⋊C4 C2×Dic22 C2×C4×D11 C44 D22 C22 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 2 1 1 1 2 2 2 5 15 20 5 5

Matrix representation of D222Q8 in GL6(𝔽89)

 88 0 0 0 0 0 0 88 0 0 0 0 0 0 51 51 0 0 0 0 38 45 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 88 0 0 0 0 0 75 1 0 0 0 0 0 0 51 51 0 0 0 0 45 38 0 0 0 0 0 0 88 0 0 0 0 0 0 88
,
 83 39 0 0 0 0 63 6 0 0 0 0 0 0 88 0 0 0 0 0 42 1 0 0 0 0 0 0 0 88 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 41 86 0 0 0 0 86 48

`G:=sub<GL(6,GF(89))| [88,0,0,0,0,0,0,88,0,0,0,0,0,0,51,38,0,0,0,0,51,45,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[88,75,0,0,0,0,0,1,0,0,0,0,0,0,51,45,0,0,0,0,51,38,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[83,63,0,0,0,0,39,6,0,0,0,0,0,0,88,42,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,88,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,86,0,0,0,0,86,48] >;`

D222Q8 in GAP, Magma, Sage, TeX

`D_{22}\rtimes_2Q_8`
`% in TeX`

`G:=Group("D22:2Q8");`
`// GroupNames label`

`G:=SmallGroup(352,92);`
`// by ID`

`G=gap.SmallGroup(352,92);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,122,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^22=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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