Copied to
clipboard

G = D887C2order 352 = 25·11

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D887C2, C44.35D4, C8.17D22, C4.20D44, Dic447C2, C22.1D44, C88.17C22, C44.30C23, D44.7C22, Dic22.6C22, (C2×C88)⋊6C2, (C2×C8)⋊4D11, C111(C4○D8), C8⋊D117C2, (C2×C22).18D4, C22.11(C2×D4), C2.13(C2×D44), (C2×C4).81D22, D445C21C2, (C2×C44).99C22, C4.28(C22×D11), SmallGroup(352,99)

Series: Derived Chief Lower central Upper central

C1C44 — D887C2
C1C11C22C44D44D445C2 — D887C2
C11C22C44 — D887C2
C1C4C2×C4C2×C8

Generators and relations for D887C2
 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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C11, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D11 [×2], C22, C22, C4○D8, Dic11 [×2], C44 [×2], D22 [×2], C2×C22, C88 [×2], Dic22 [×2], C4×D11 [×2], D44 [×2], C11⋊D4 [×2], C2×C44, C8⋊D11 [×2], D88, Dic44, C2×C88, D445C2 [×2], D887C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D11, C4○D8, D22 [×3], D44 [×2], C22×D11, C2×D44, D887C2

Smallest permutation representation of D887C2
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)(89 121)(90 120)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(122 176)(123 175)(124 174)(125 173)(126 172)(127 171)(128 170)(129 169)(130 168)(131 167)(132 166)(133 165)(134 164)(135 163)(136 162)(137 161)(138 160)(139 159)(140 158)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 170)(28 171)(29 172)(30 173)(31 174)(32 175)(33 176)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)

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

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

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

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○D8D22D22D44D44D887C2
kernelD887C2C8⋊D11D88Dic44C2×C88D445C2C44C2×C22C2×C8C11C8C2×C4C4C22C1
# reps1211121154105101040

Matrix representation of D887C2 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] >;

D887C2 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

׿
×
𝔽