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## G = D88⋊5C2order 352 = 25·11

### 5th semidirect product of D88 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D88⋊5C2
 Chief series C1 — C11 — C22 — C44 — C4×D11 — D44⋊C2 — D88⋊5C2
 Lower central C11 — C22 — C44 — D88⋊5C2
 Upper central C1 — C2 — C4 — Q16

Generators and relations for D885C2
G = < a,b,c | a88=b2=c2=1, bab=a-1, cac=a65, cbc=a20b >

Subgroups: 442 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, C11, C2×C8, D8, SD16, Q16, C4○D4, D11, C22, C4○D8, Dic11, C44, C44, D22, D22, C11⋊C8, C88, C4×D11, C4×D11, D44, D44, Q8×C11, C8×D11, D88, Q8⋊D11, C11×Q16, D44⋊C2, D885C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4○D8, D22, C22×D11, D4×D11, D885C2

Smallest permutation representation of D885C2
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 88)(2 87)(3 86)(4 85)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(127 176)(128 175)(129 174)(130 173)(131 172)(132 171)(133 170)(134 169)(135 168)(136 167)(137 166)(138 165)(139 164)(140 163)(141 162)(142 161)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)
(1 174)(2 151)(3 128)(4 105)(5 170)(6 147)(7 124)(8 101)(9 166)(10 143)(11 120)(12 97)(13 162)(14 139)(15 116)(16 93)(17 158)(18 135)(19 112)(20 89)(21 154)(22 131)(23 108)(24 173)(25 150)(26 127)(27 104)(28 169)(29 146)(30 123)(31 100)(32 165)(33 142)(34 119)(35 96)(36 161)(37 138)(38 115)(39 92)(40 157)(41 134)(42 111)(43 176)(44 153)(45 130)(46 107)(47 172)(48 149)(49 126)(50 103)(51 168)(52 145)(53 122)(54 99)(55 164)(56 141)(57 118)(58 95)(59 160)(60 137)(61 114)(62 91)(63 156)(64 133)(65 110)(66 175)(67 152)(68 129)(69 106)(70 171)(71 148)(72 125)(73 102)(74 167)(75 144)(76 121)(77 98)(78 163)(79 140)(80 117)(81 94)(82 159)(83 136)(84 113)(85 90)(86 155)(87 132)(88 109)```

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

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

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

49 conjugacy classes

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

49 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D11 C4○D8 D22 D22 D4×D11 D88⋊5C2 kernel D88⋊5C2 C8×D11 D88 Q8⋊D11 C11×Q16 D44⋊C2 Dic11 D22 Q16 C11 C8 Q8 C2 C1 # reps 1 1 1 2 1 2 1 1 5 4 5 10 5 10

Matrix representation of D885C2 in GL4(𝔽89) generated by

 78 13 0 0 60 18 0 0 0 0 64 67 0 0 85 0
,
 53 2 0 0 20 36 0 0 0 0 64 67 0 0 85 25
,
 33 1 0 0 69 56 0 0 0 0 55 79 0 0 71 34
`G:=sub<GL(4,GF(89))| [78,60,0,0,13,18,0,0,0,0,64,85,0,0,67,0],[53,20,0,0,2,36,0,0,0,0,64,85,0,0,67,25],[33,69,0,0,1,56,0,0,0,0,55,71,0,0,79,34] >;`

D885C2 in GAP, Magma, Sage, TeX

`D_{88}\rtimes_5C_2`
`% in TeX`

`G:=Group("D88:5C2");`
`// GroupNames label`

`G:=SmallGroup(352,114);`
`// by ID`

`G=gap.SmallGroup(352,114);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,362,116,86,297,159,69,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^88=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^65,c*b*c=a^20*b>;`
`// generators/relations`

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