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## G = C44.17D4order 352 = 25·11

### 17th non-split extension by C44 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C44.17D4
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C4×Dic11 — C44.17D4
 Lower central C11 — C2×C22 — C44.17D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C44.17D4
G = < a,b,c | a44=b4=1, c2=a22, bab-1=a21, cac-1=a-1, cbc-1=a22b-1 >

Subgroups: 346 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C11, C42, C22⋊C4, C2×D4, C2×Q8, C22, C22, C22, C4.4D4, Dic11, C44, C2×C22, C2×C22, Dic22, C2×Dic11, C2×C44, D4×C11, C22×C22, C4×Dic11, C23.D11, C2×Dic22, D4×C22, C44.17D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4.4D4, D22, C11⋊D4, C22×D11, D42D11, C2×C11⋊D4, C44.17D4

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

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 D4 C4○D4 D11 D22 D22 C11⋊D4 D4⋊2D11 kernel C44.17D4 C4×Dic11 C23.D11 C2×Dic22 D4×C22 C44 C22 C2×D4 C2×C4 C23 C4 C2 # reps 1 1 4 1 1 2 4 5 5 10 20 10

Matrix representation of C44.17D4 in GL4(𝔽89) generated by

 22 0 0 0 31 85 0 0 0 0 1 3 0 0 29 88
,
 68 9 0 0 30 21 0 0 0 0 34 13 0 0 7 55
,
 68 9 0 0 50 21 0 0 0 0 34 0 0 0 7 55
`G:=sub<GL(4,GF(89))| [22,31,0,0,0,85,0,0,0,0,1,29,0,0,3,88],[68,30,0,0,9,21,0,0,0,0,34,7,0,0,13,55],[68,50,0,0,9,21,0,0,0,0,34,7,0,0,0,55] >;`

C44.17D4 in GAP, Magma, Sage, TeX

`C_{44}._{17}D_4`
`% in TeX`

`G:=Group("C44.17D4");`
`// GroupNames label`

`G:=SmallGroup(352,131);`
`// by ID`

`G=gap.SmallGroup(352,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,55,506,116,11525]);`
`// Polycyclic`

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

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