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## G = C44.48D4order 352 = 25·11

### 5th non-split extension by C44 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C44.48D4
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C2×Dic22 — C44.48D4
 Lower central C11 — C2×C22 — C44.48D4
 Upper central C1 — C22 — C22×C4

Generators and relations for C44.48D4
G = < a,b,c | a44=b4=1, c2=a22, bab-1=cac-1=a-1, cbc-1=a22b-1 >

Subgroups: 330 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C22, C22, C22⋊Q8, Dic11, C44, C44, C2×C22, C2×C22, C2×C22, Dic22, C2×Dic11, C2×C44, C2×C44, C22×C22, Dic11⋊C4, C44⋊C4, C23.D11, C2×Dic22, C22×C44, C44.48D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, D11, C22⋊Q8, D22, Dic22, C11⋊D4, C22×D11, C2×Dic22, D445C2, C2×C11⋊D4, C44.48D4

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

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

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

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

94 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 11A ··· 11E 22A ··· 22AI 44A ··· 44AN order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 2 2 2 2 44 44 44 44 2 ··· 2 2 ··· 2 2 ··· 2

94 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 D4 Q8 C4○D4 D11 D22 D22 C11⋊D4 Dic22 D44⋊5C2 kernel C44.48D4 Dic11⋊C4 C44⋊C4 C23.D11 C2×Dic22 C22×C44 C44 C2×C22 C22 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 1 2 1 1 2 2 2 5 10 5 20 20 20

Matrix representation of C44.48D4 in GL4(𝔽89) generated by

 44 0 0 0 0 87 0 0 0 0 79 0 0 0 0 80
,
 0 1 0 0 88 0 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 88 0
`G:=sub<GL(4,GF(89))| [44,0,0,0,0,87,0,0,0,0,79,0,0,0,0,80],[0,88,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,88,0,0,1,0] >;`

C44.48D4 in GAP, Magma, Sage, TeX

`C_{44}._{48}D_4`
`% in TeX`

`G:=Group("C44.48D4");`
`// GroupNames label`

`G:=SmallGroup(352,119);`
`// by ID`

`G=gap.SmallGroup(352,119);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,217,103,218,11525]);`
`// Polycyclic`

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

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