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## G = C44.23D4order 352 = 25·11

### 23rd non-split extension by C44 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C44.23D4
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×D44 — C44.23D4
 Lower central C11 — C2×C22 — C44.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C44.23D4
G = < a,b,c | a44=b4=c2=1, bab-1=a21, cac=a-1, cbc=a22b-1 >

Subgroups: 506 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C11, C42, C22⋊C4, C2×D4, C2×Q8, D11, C22, C22, C4.4D4, Dic11, C44, C44, D22, C2×C22, D44, C2×Dic11, C2×C44, C2×C44, Q8×C11, C22×D11, C4×Dic11, D22⋊C4, C2×D44, Q8×C22, C44.23D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4.4D4, D22, C11⋊D4, C22×D11, D44⋊C2, C2×C11⋊D4, C44.23D4

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 11A ··· 11E 22A ··· 22O 44A ··· 44AD 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 44 44 2 2 4 4 22 22 22 22 2 ··· 2 2 ··· 2 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 C4○D4 D11 D22 C11⋊D4 D44⋊C2 kernel C44.23D4 C4×Dic11 D22⋊C4 C2×D44 Q8×C22 C44 C22 C2×Q8 C2×C4 C4 C2 # reps 1 1 4 1 1 2 4 5 15 20 10

Matrix representation of C44.23D4 in GL4(𝔽89) generated by

 3 8 0 0 81 68 0 0 0 0 31 2 0 0 53 58
,
 42 28 0 0 80 47 0 0 0 0 55 0 0 0 0 55
,
 1 0 0 0 86 88 0 0 0 0 1 0 0 0 58 88
`G:=sub<GL(4,GF(89))| [3,81,0,0,8,68,0,0,0,0,31,53,0,0,2,58],[42,80,0,0,28,47,0,0,0,0,55,0,0,0,0,55],[1,86,0,0,0,88,0,0,0,0,1,58,0,0,0,88] >;`

C44.23D4 in GAP, Magma, Sage, TeX

`C_{44}._{23}D_4`
`% in TeX`

`G:=Group("C44.23D4");`
`// GroupNames label`

`G:=SmallGroup(352,142);`
`// by ID`

`G=gap.SmallGroup(352,142);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,218,188,86,11525]);`
`// 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=a^22*b^-1>;`
`// generators/relations`

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