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## G = C44.53D4order 352 = 25·11

### 10th non-split extension by C44 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — C44.53D4
 Chief series C1 — C11 — C22 — C44 — C2×C44 — C2×C11⋊C8 — C44.53D4
 Lower central C11 — C22 — C44 — C44.53D4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C44.53D4
G = < a,b,c | a44=1, b4=a22, c2=a33, bab-1=cac-1=a21, cbc-1=a22b3 >

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

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + - + + - image C1 C2 C2 C2 C4 D4 Q8 D11 C8.C4 D22 C4×D11 C11⋊D4 Dic22 C44.53D4 kernel C44.53D4 C2×C11⋊C8 C44.C4 C11×M4(2) C11⋊C8 C44 C2×C22 M4(2) C11 C2×C4 C4 C4 C22 C1 # reps 1 1 1 1 4 1 1 5 4 5 10 10 10 10

Matrix representation of C44.53D4 in GL4(𝔽89) generated by

 55 0 0 0 0 55 0 0 0 0 5 81 0 0 81 13
,
 37 0 0 0 0 77 0 0 0 0 33 9 0 0 47 56
,
 0 77 0 0 77 0 0 0 0 0 21 13 0 0 55 68
`G:=sub<GL(4,GF(89))| [55,0,0,0,0,55,0,0,0,0,5,81,0,0,81,13],[37,0,0,0,0,77,0,0,0,0,33,47,0,0,9,56],[0,77,0,0,77,0,0,0,0,0,21,55,0,0,13,68] >;`

C44.53D4 in GAP, Magma, Sage, TeX

`C_{44}._{53}D_4`
`% in TeX`

`G:=Group("C44.53D4");`
`// GroupNames label`

`G:=SmallGroup(352,28);`
`// by ID`

`G=gap.SmallGroup(352,28);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,121,31,86,297,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^44=1,b^4=a^22,c^2=a^33,b*a*b^-1=c*a*c^-1=a^21,c*b*c^-1=a^22*b^3>;`
`// generators/relations`

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