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## G = C44.47D4order 352 = 25·11

### 4th non-split extension by C44 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C44.47D4
 Chief series C1 — C11 — C22 — C44 — C2×C44 — C2×Dic22 — C44.47D4
 Lower central C11 — C22 — C2×C22 — C44.47D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C44.47D4
G = < a,b,c | a44=1, b4=c2=a22, bab-1=cac-1=a-1, cbc-1=a33b3 >

Smallest permutation representation of C44.47D4
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)
(1 151 34 162 23 173 12 140)(2 150 35 161 24 172 13 139)(3 149 36 160 25 171 14 138)(4 148 37 159 26 170 15 137)(5 147 38 158 27 169 16 136)(6 146 39 157 28 168 17 135)(7 145 40 156 29 167 18 134)(8 144 41 155 30 166 19 133)(9 143 42 154 31 165 20 176)(10 142 43 153 32 164 21 175)(11 141 44 152 33 163 22 174)(45 128 56 117 67 106 78 95)(46 127 57 116 68 105 79 94)(47 126 58 115 69 104 80 93)(48 125 59 114 70 103 81 92)(49 124 60 113 71 102 82 91)(50 123 61 112 72 101 83 90)(51 122 62 111 73 100 84 89)(52 121 63 110 74 99 85 132)(53 120 64 109 75 98 86 131)(54 119 65 108 76 97 87 130)(55 118 66 107 77 96 88 129)
(1 132 23 110)(2 131 24 109)(3 130 25 108)(4 129 26 107)(5 128 27 106)(6 127 28 105)(7 126 29 104)(8 125 30 103)(9 124 31 102)(10 123 32 101)(11 122 33 100)(12 121 34 99)(13 120 35 98)(14 119 36 97)(15 118 37 96)(16 117 38 95)(17 116 39 94)(18 115 40 93)(19 114 41 92)(20 113 42 91)(21 112 43 90)(22 111 44 89)(45 158 67 136)(46 157 68 135)(47 156 69 134)(48 155 70 133)(49 154 71 176)(50 153 72 175)(51 152 73 174)(52 151 74 173)(53 150 75 172)(54 149 76 171)(55 148 77 170)(56 147 78 169)(57 146 79 168)(58 145 80 167)(59 144 81 166)(60 143 82 165)(61 142 83 164)(62 141 84 163)(63 140 85 162)(64 139 86 161)(65 138 87 160)(66 137 88 159)```

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 D4 D11 D22 D44 C11⋊D4 C4×D11 C4.10D4 C44.47D4 kernel C44.47D4 C44.C4 C11×M4(2) C2×Dic22 C2×Dic11 C44 M4(2) C2×C4 C4 C4 C22 C11 C1 # reps 1 1 1 1 4 2 5 5 10 10 10 1 10

Matrix representation of C44.47D4 in GL4(𝔽89) generated by

 43 65 0 0 73 40 0 0 47 61 58 65 2 69 62 25
,
 30 39 52 12 88 18 70 74 2 62 40 64 37 70 18 1
,
 71 47 0 0 48 18 0 0 81 25 75 15 75 54 64 14
`G:=sub<GL(4,GF(89))| [43,73,47,2,65,40,61,69,0,0,58,62,0,0,65,25],[30,88,2,37,39,18,62,70,52,70,40,18,12,74,64,1],[71,48,81,75,47,18,25,54,0,0,75,64,0,0,15,14] >;`

C44.47D4 in GAP, Magma, Sage, TeX

`C_{44}._{47}D_4`
`% in TeX`

`G:=Group("C44.47D4");`
`// GroupNames label`

`G:=SmallGroup(352,30);`
`// by ID`

`G=gap.SmallGroup(352,30);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,121,31,362,86,297,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^44=1,b^4=c^2=a^22,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^33*b^3>;`
`// generators/relations`

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