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G = D88⋊7C2  order 352 = 25·11

The semidirect product of D88 and C2 acting through Inn(D88)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D88⋊7C2, C44.35D4, C8.17D22, C4.20D44, Dic44⋊7C2, C22.1D44, C88.17C22, C44.30C23, D44.7C22, Dic22.6C22, (C2×C88)⋊6C2, (C2×C8)⋊4D11, C11⋊1(C4○D8), C8⋊D11⋊7C2, (C2×C22).18D4, C22.11(C2×D4), C2.13(C2×D44), (C2×C4).81D22, D44⋊5C2⋊1C2, (C2×C44).99C22, C4.28(C22×D11), SmallGroup(352,99)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C44 — D88⋊7C2
Chief series
C1 — C11 — C22 — C44 — D44 — D44⋊5C2 — D88⋊7C2
Lower central
C11 — C22 — C44 — D88⋊7C2
Upper central
C1 — C4 — C2×C4 — C2×C8

Generators and relations for D88⋊7C2
 G = < a,b,c | a88=b2=c2=1, bab=a-1, ac=ca, cbc=a44b >

Subgroups: 442 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C11, C2×C8, D8, SD16, Q16, C4○D4, D11, C22, C22, C4○D8, Dic11, C44, D22, C2×C22, C88, Dic22, C4×D11, D44, C11⋊D4, C2×C44, C8⋊D11, D88, Dic44, C2×C88, D44⋊5C2, D88⋊7C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4○D8, D22, D44, C22×D11, C2×D44, D88⋊7C2

Smallest permutation representation of D88⋊7C2
►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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(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)(56 88)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(89 97)(90 96)(91 95)(92 94)(98 176)(99 175)(100 174)(101 173)(102 172)(103 171)(104 170)(105 169)(106 168)(107 167)(108 166)(109 165)(110 164)(111 163)(112 162)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 154)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)

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

94 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D11A···11E22A···22O44A···44T88A···88AN
order1222244444888811···1122···2244···4488···88
size1124444112444422222···22···22···22···2

94 irreducible representations

dim111111222222222
type+++++++++++++
imageC1C2C2C2C2C2D4D4D11C4○D8D22D22D44D44D88⋊7C2
kernelD88⋊7C2C8⋊D11D88Dic44C2×C88D44⋊5C2C44C2×C22C2×C8C11C8C2×C4C4C22C1
# reps1211121154105101040

Matrix representation of D88⋊7C2 ►in GL2(𝔽89) generated by

6712
1453
,
4073
6149
,
2848
5661
G:=sub<GL(2,GF(89))| [67,14,12,53],[40,61,73,49],[28,56,48,61] >;
 

D88⋊7C2 in GAP, Magma, Sage, TeX 

D_{88}\rtimes_7C_2
 
% in TeX
 
G:=Group("D88:7C2");
 
// GroupNames label
 
G:=SmallGroup(352,99);
 
// by ID
 
G=gap.SmallGroup(352,99);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,50,579,69,11525]);
 
// Polycyclic
 
G:=Group<a,b,c|a^88=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^44*b>;
 
// generators/relations
 

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