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## G = C4⋊C4⋊D11order 352 = 25·11

### 6th semidirect product of C4⋊C4 and D11 acting via D11/C11=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊C4⋊D11
G = < a,b,c,d | a4=b4=c11=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 354 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C11, C42, C22⋊C4, C4⋊C4, C4⋊C4, D11, C22, C422C2, Dic11, C44, D22, C2×C22, C2×Dic11, C2×C44, C22×D11, C4×Dic11, Dic11⋊C4, C44⋊C4, D22⋊C4, C11×C4⋊C4, C4⋊C4⋊D11
Quotients: C1, C2, C22, C23, C4○D4, D11, C422C2, D22, C22×D11, D445C2, D42D11, D44⋊C2, C4⋊C4⋊D11

Smallest permutation representation of C4⋊C4⋊D11
On 176 points
Generators in S176
```(1 87 21 76)(2 88 22 77)(3 78 12 67)(4 79 13 68)(5 80 14 69)(6 81 15 70)(7 82 16 71)(8 83 17 72)(9 84 18 73)(10 85 19 74)(11 86 20 75)(23 56 34 45)(24 57 35 46)(25 58 36 47)(26 59 37 48)(27 60 38 49)(28 61 39 50)(29 62 40 51)(30 63 41 52)(31 64 42 53)(32 65 43 54)(33 66 44 55)(89 155 100 166)(90 156 101 167)(91 157 102 168)(92 158 103 169)(93 159 104 170)(94 160 105 171)(95 161 106 172)(96 162 107 173)(97 163 108 174)(98 164 109 175)(99 165 110 176)(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)
(1 120 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 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 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 77)(53 76)(54 75)(55 74)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 88)(64 87)(65 86)(66 85)(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)(133 172)(134 171)(135 170)(136 169)(137 168)(138 167)(139 166)(140 176)(141 175)(142 174)(143 173)(144 161)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 165)(152 164)(153 163)(154 162)```

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4○D4 D11 D22 D44⋊5C2 D4⋊2D11 D44⋊C2 kernel C4⋊C4⋊D11 C4×Dic11 Dic11⋊C4 C44⋊C4 D22⋊C4 C11×C4⋊C4 C22 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 1 1 1 3 1 6 5 15 20 5 5

Matrix representation of C4⋊C4⋊D11 in GL4(𝔽89) generated by

 42 28 0 0 61 47 0 0 0 0 88 2 0 0 88 1
,
 55 0 0 0 0 55 0 0 0 0 55 68 0 0 0 34
,
 0 1 0 0 88 86 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 88
`G:=sub<GL(4,GF(89))| [42,61,0,0,28,47,0,0,0,0,88,88,0,0,2,1],[55,0,0,0,0,55,0,0,0,0,55,0,0,0,68,34],[0,88,0,0,1,86,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,88] >;`

C4⋊C4⋊D11 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes D_{11}`
`% in TeX`

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

`G:=SmallGroup(352,93);`
`// by ID`

`G=gap.SmallGroup(352,93);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,55,218,188,86,11525]);`
`// Polycyclic`

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

׿
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