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G = D44⋊C4order 352 = 25·11

5th semidirect product of D44 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D44⋊C4
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×D44 — D44⋊C4
 Lower central C11 — C22 — D44⋊C4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D44⋊C4
G = < a,b,c | a44=b2=c4=1, bab=a-1, cac-1=a23, cbc-1=a22b >

Subgroups: 594 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C4×D4, Dic11, Dic11, C44, C44, D22, D22, C2×C22, C4×D11, D44, C2×Dic11, C2×C44, C2×C44, C22×D11, C4×Dic11, D22⋊C4, C11×C4⋊C4, C2×C4×D11, C2×D44, D44⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D11, C4×D4, D22, C4×D11, C22×D11, C2×C4×D11, D4×D11, D44⋊C2, D44⋊C4

Smallest permutation representation of D44⋊C4
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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(46 88)(47 87)(48 86)(49 85)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(89 113)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)(121 125)(122 124)(133 143)(134 142)(135 141)(136 140)(137 139)(144 176)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)
(1 73 155 129)(2 52 156 108)(3 75 157 131)(4 54 158 110)(5 77 159 89)(6 56 160 112)(7 79 161 91)(8 58 162 114)(9 81 163 93)(10 60 164 116)(11 83 165 95)(12 62 166 118)(13 85 167 97)(14 64 168 120)(15 87 169 99)(16 66 170 122)(17 45 171 101)(18 68 172 124)(19 47 173 103)(20 70 174 126)(21 49 175 105)(22 72 176 128)(23 51 133 107)(24 74 134 130)(25 53 135 109)(26 76 136 132)(27 55 137 111)(28 78 138 90)(29 57 139 113)(30 80 140 92)(31 59 141 115)(32 82 142 94)(33 61 143 117)(34 84 144 96)(35 63 145 119)(36 86 146 98)(37 65 147 121)(38 88 148 100)(39 67 149 123)(40 46 150 102)(41 69 151 125)(42 48 152 104)(43 71 153 127)(44 50 154 106)```

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G 4H 4I 4J 4K 4L 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 22 22 22 22 2 ··· 2 11 11 11 11 22 22 2 ··· 2 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 D11 D22 C4×D11 D4×D11 D44⋊C2 kernel D44⋊C4 C4×Dic11 D22⋊C4 C11×C4⋊C4 C2×C4×D11 C2×D44 D44 Dic11 C22 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 2 1 2 1 8 2 2 5 15 20 5 5

Matrix representation of D44⋊C4 in GL4(𝔽89) generated by

 13 8 0 0 8 5 0 0 0 0 57 21 0 0 19 32
,
 55 68 0 0 55 34 0 0 0 0 88 0 0 0 69 1
,
 34 0 0 0 0 34 0 0 0 0 20 87 0 0 67 69
`G:=sub<GL(4,GF(89))| [13,8,0,0,8,5,0,0,0,0,57,19,0,0,21,32],[55,55,0,0,68,34,0,0,0,0,88,69,0,0,0,1],[34,0,0,0,0,34,0,0,0,0,20,67,0,0,87,69] >;`

D44⋊C4 in GAP, Magma, Sage, TeX

`D_{44}\rtimes C_4`
`% in TeX`

`G:=Group("D44:C4");`
`// GroupNames label`

`G:=SmallGroup(352,88);`
`// by ID`

`G=gap.SmallGroup(352,88);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,188,50,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^44=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^23,c*b*c^-1=a^22*b>;`
`// generators/relations`

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