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## G = C42⋊2D11order 352 = 25·11

### 2nd semidirect product of C42 and D11 acting via D11/C11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C42⋊2D11
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — D22⋊C4 — C42⋊2D11
 Lower central C11 — C2×C22 — C42⋊2D11
 Upper central C1 — C22 — C42

Generators and relations for C422D11
G = < a,b,c,d | a4=b4=c11=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >

Subgroups: 354 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C11, C42, C22⋊C4, C4⋊C4, D11, C22, C422C2, Dic11, C44, D22, C2×C22, C2×Dic11, C2×C44, C22×D11, Dic11⋊C4, D22⋊C4, C4×C44, C422D11
Quotients: C1, C2, C22, C23, C4○D4, D11, C422C2, D22, C22×D11, D445C2, C422D11

Smallest permutation representation of C422D11
On 176 points
Generators in S176
```(1 120 32 98)(2 121 33 99)(3 111 23 89)(4 112 24 90)(5 113 25 91)(6 114 26 92)(7 115 27 93)(8 116 28 94)(9 117 29 95)(10 118 30 96)(11 119 31 97)(12 122 34 100)(13 123 35 101)(14 124 36 102)(15 125 37 103)(16 126 38 104)(17 127 39 105)(18 128 40 106)(19 129 41 107)(20 130 42 108)(21 131 43 109)(22 132 44 110)(45 155 67 133)(46 156 68 134)(47 157 69 135)(48 158 70 136)(49 159 71 137)(50 160 72 138)(51 161 73 139)(52 162 74 140)(53 163 75 141)(54 164 76 142)(55 165 77 143)(56 166 78 144)(57 167 79 145)(58 168 80 146)(59 169 81 147)(60 170 82 148)(61 171 83 149)(62 172 84 150)(63 173 85 151)(64 174 86 152)(65 175 87 153)(66 176 88 154)
(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)(89 144 100 133)(90 145 101 134)(91 146 102 135)(92 147 103 136)(93 148 104 137)(94 149 105 138)(95 150 106 139)(96 151 107 140)(97 152 108 141)(98 153 109 142)(99 154 110 143)(111 166 122 155)(112 167 123 156)(113 168 124 157)(114 169 125 158)(115 170 126 159)(116 171 127 160)(117 172 128 161)(118 173 129 162)(119 174 130 163)(120 175 131 164)(121 176 132 165)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 88)(53 87)(54 86)(55 85)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 77)(64 76)(65 75)(66 74)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 110)(97 109)(98 108)(99 107)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 132)(119 131)(120 130)(121 129)(133 161)(134 160)(135 159)(136 158)(137 157)(138 156)(139 155)(140 165)(141 164)(142 163)(143 162)(144 172)(145 171)(146 170)(147 169)(148 168)(149 167)(150 166)(151 176)(152 175)(153 174)(154 173)```

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

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

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

94 conjugacy classes

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

94 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4○D4 D11 D22 D44⋊5C2 kernel C42⋊2D11 Dic11⋊C4 D22⋊C4 C4×C44 C22 C42 C2×C4 C2 # reps 1 3 3 1 6 5 15 60

Matrix representation of C422D11 in GL4(𝔽89) generated by

 55 0 0 0 0 55 0 0 0 0 29 59 0 0 34 60
,
 86 39 0 0 50 3 0 0 0 0 34 0 0 0 0 34
,
 0 1 0 0 88 7 0 0 0 0 82 1 0 0 76 78
,
 0 1 0 0 1 0 0 0 0 0 78 88 0 0 31 11
`G:=sub<GL(4,GF(89))| [55,0,0,0,0,55,0,0,0,0,29,34,0,0,59,60],[86,50,0,0,39,3,0,0,0,0,34,0,0,0,0,34],[0,88,0,0,1,7,0,0,0,0,82,76,0,0,1,78],[0,1,0,0,1,0,0,0,0,0,78,31,0,0,88,11] >;`

C422D11 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2D_{11}`
`% in TeX`

`G:=Group("C4^2:2D11");`
`// GroupNames label`

`G:=SmallGroup(352,71);`
`// by ID`

`G=gap.SmallGroup(352,71);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,55,506,86,11525]);`
`// Polycyclic`

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

׿
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