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## G = Q16⋊D11order 352 = 25·11

### 2nd semidirect product of Q16 and D11 acting via D11/C11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — Q16⋊D11
 Chief series C1 — C11 — C22 — C44 — C4×D11 — Q8×D11 — Q16⋊D11
 Lower central C11 — C22 — C44 — Q16⋊D11
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊D11
G = < a,b,c,d | a8=c11=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 394 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, Q8, C11, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, D11, C22, C8.C22, Dic11, Dic11, C44, C44, D22, D22, C11⋊C8, C88, Dic22, Dic22, C4×D11, C4×D11, D44, D44, Q8×C11, C88⋊C2, C8⋊D11, Q8⋊D11, C11⋊Q16, C11×Q16, Q8×D11, D44⋊C2, Q16⋊D11
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8.C22, D22, C22×D11, D4×D11, Q16⋊D11

Smallest permutation representation of Q16⋊D11
On 176 points
Generators in S176
```(1 164 32 153 21 175 43 142)(2 165 33 154 22 176 44 143)(3 155 23 144 12 166 34 133)(4 156 24 145 13 167 35 134)(5 157 25 146 14 168 36 135)(6 158 26 147 15 169 37 136)(7 159 27 148 16 170 38 137)(8 160 28 149 17 171 39 138)(9 161 29 150 18 172 40 139)(10 162 30 151 19 173 41 140)(11 163 31 152 20 174 42 141)(45 111 78 89 56 122 67 100)(46 112 79 90 57 123 68 101)(47 113 80 91 58 124 69 102)(48 114 81 92 59 125 70 103)(49 115 82 93 60 126 71 104)(50 116 83 94 61 127 72 105)(51 117 84 95 62 128 73 106)(52 118 85 96 63 129 74 107)(53 119 86 97 64 130 75 108)(54 120 87 98 65 131 76 109)(55 121 88 99 66 132 77 110)
(1 65 21 54)(2 66 22 55)(3 56 12 45)(4 57 13 46)(5 58 14 47)(6 59 15 48)(7 60 16 49)(8 61 17 50)(9 62 18 51)(10 63 19 52)(11 64 20 53)(23 78 34 67)(24 79 35 68)(25 80 36 69)(26 81 37 70)(27 82 38 71)(28 83 39 72)(29 84 40 73)(30 85 41 74)(31 86 42 75)(32 87 43 76)(33 88 44 77)(89 166 100 155)(90 167 101 156)(91 168 102 157)(92 169 103 158)(93 170 104 159)(94 171 105 160)(95 172 106 161)(96 173 107 162)(97 174 108 163)(98 175 109 164)(99 176 110 165)(111 133 122 144)(112 134 123 145)(113 135 124 146)(114 136 125 147)(115 137 126 148)(116 138 127 149)(117 139 128 150)(118 140 129 151)(119 141 130 152)(120 142 131 153)(121 143 132 154)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 55)(11 54)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 66)(20 65)(21 64)(22 63)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 88)(31 87)(32 86)(33 85)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 77)(42 76)(43 75)(44 74)(89 139)(90 138)(91 137)(92 136)(93 135)(94 134)(95 133)(96 143)(97 142)(98 141)(99 140)(100 150)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 154)(108 153)(109 152)(110 151)(111 172)(112 171)(113 170)(114 169)(115 168)(116 167)(117 166)(118 176)(119 175)(120 174)(121 173)(122 161)(123 160)(124 159)(125 158)(126 157)(127 156)(128 155)(129 165)(130 164)(131 163)(132 162)```

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

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

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

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 C8.C22 D4×D11 Q16⋊D11 kernel Q16⋊D11 C88⋊C2 C8⋊D11 Q8⋊D11 C11⋊Q16 C11×Q16 Q8×D11 D44⋊C2 Dic11 D22 Q16 C8 Q8 C11 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 5 5 10 1 5 10

Matrix representation of Q16⋊D11 in GL4(𝔽89) generated by

 0 0 23 13 0 0 55 39 57 70 29 63 36 61 22 60
,
 80 27 25 3 72 52 82 15 2 29 36 62 20 24 16 10
,
 85 1 0 0 62 51 0 0 0 0 84 1 0 0 6 52
,
 16 58 82 5 43 73 23 2 44 68 9 47 28 24 53 80
`G:=sub<GL(4,GF(89))| [0,0,57,36,0,0,70,61,23,55,29,22,13,39,63,60],[80,72,2,20,27,52,29,24,25,82,36,16,3,15,62,10],[85,62,0,0,1,51,0,0,0,0,84,6,0,0,1,52],[16,43,44,28,58,73,68,24,82,23,9,53,5,2,47,80] >;`

Q16⋊D11 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_{11}`
`% in TeX`

`G:=Group("Q16:D11");`
`// GroupNames label`

`G:=SmallGroup(352,113);`
`// by ID`

`G=gap.SmallGroup(352,113);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,362,116,86,297,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^11=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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