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## G = C44⋊D4order 352 = 25·11

### 3rd semidirect product of C44 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C44⋊D4
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×D44 — C44⋊D4
 Lower central C11 — C2×C22 — C44⋊D4
 Upper central C1 — C22 — C2×D4

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

Subgroups: 730 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C42, C2×D4, C2×D4, D11, C22, C22, C22, C41D4, Dic11, C44, D22, C2×C22, C2×C22, D44, C2×Dic11, C11⋊D4, C2×C44, D4×C11, C22×D11, C22×C22, C4×Dic11, C2×D44, C2×C11⋊D4, D4×C22, C44⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C41D4, D22, C11⋊D4, C22×D11, D4×D11, C2×C11⋊D4, C44⋊D4

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 4 4 44 44 2 2 22 22 22 22 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 D4 D11 D22 D22 C11⋊D4 D4×D11 kernel C44⋊D4 C4×Dic11 C2×D44 C2×C11⋊D4 D4×C22 Dic11 C44 C2×D4 C2×C4 C23 C4 C2 # reps 1 1 1 4 1 4 2 5 5 10 20 10

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

 0 1 0 0 88 0 0 0 0 0 2 56 0 0 24 5
,
 88 0 0 0 0 88 0 0 0 0 14 72 0 0 43 75
,
 1 0 0 0 0 88 0 0 0 0 88 11 0 0 0 1
`G:=sub<GL(4,GF(89))| [0,88,0,0,1,0,0,0,0,0,2,24,0,0,56,5],[88,0,0,0,0,88,0,0,0,0,14,43,0,0,72,75],[1,0,0,0,0,88,0,0,0,0,88,0,0,0,11,1] >;`

C44⋊D4 in GAP, Magma, Sage, TeX

`C_{44}\rtimes D_4`
`% in TeX`

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

`G:=SmallGroup(352,135);`
`// by ID`

`G=gap.SmallGroup(352,135);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,218,188,11525]);`
`// Polycyclic`

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

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