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## G = D44.2C4order 352 = 25·11

### The non-split extension by D44 of C4 acting through Inn(D44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D44.2C4
 Chief series C1 — C11 — C22 — C44 — C4×D11 — D44⋊5C2 — D44.2C4
 Lower central C11 — C22 — D44.2C4
 Upper central C1 — C8 — C2×C8

Generators and relations for D44.2C4
G = < a,b,c | a44=b2=1, c4=a22, bab=a-1, ac=ca, bc=cb >

Subgroups: 298 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C11, C2×C8, C2×C8, M4(2), C4○D4, D11, C22, C22, C8○D4, Dic11, C44, D22, C2×C22, C11⋊C8, C88, Dic22, C4×D11, D44, C11⋊D4, C2×C44, C8×D11, C88⋊C2, C44.C4, C2×C88, D445C2, D44.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D11, C8○D4, D22, C4×D11, C22×D11, C2×C4×D11, D44.2C4

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

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

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

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

100 conjugacy classes

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

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D11 C8○D4 D22 D22 C4×D11 C4×D11 D44.2C4 kernel D44.2C4 C8×D11 C88⋊C2 C44.C4 C2×C88 D44⋊5C2 Dic22 D44 C11⋊D4 C2×C8 C11 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 5 4 10 5 10 10 40

Matrix representation of D44.2C4 in GL2(𝔽89) generated by

 75 42 25 33
,
 3 3 27 86
,
 77 0 0 77
`G:=sub<GL(2,GF(89))| [75,25,42,33],[3,27,3,86],[77,0,0,77] >;`

D44.2C4 in GAP, Magma, Sage, TeX

`D_{44}._2C_4`
`% in TeX`

`G:=Group("D44.2C4");`
`// GroupNames label`

`G:=SmallGroup(352,96);`
`// by ID`

`G=gap.SmallGroup(352,96);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,50,69,11525]);`
`// Polycyclic`

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

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