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G = D8⋊3D11  order 352 = 25·11

The semidirect product of D8 and D11 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8⋊3D11, C8.8D22, D4.1D22, D22.1D4, Dic44⋊4C2, C44.3C23, C88.6C22, Dic11.12D4, Dic22.1C22, (C8×D11)⋊2C2, (C11×D8)⋊3C2, C11⋊2(C4○D8), D4.D11⋊2C2, C22.29(C2×D4), C2.17(D4×D11), C11⋊C8.5C22, D4⋊2D11⋊2C2, C4.3(C22×D11), (C4×D11).8C22, (D4×C11).1C22, SmallGroup(352,107)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C44 — D8⋊3D11
Chief series
C1 — C11 — C22 — C44 — C4×D11 — D4⋊2D11 — D8⋊3D11
Lower central
C11 — C22 — C44 — D8⋊3D11
Upper central
C1 — C2 — C4 — D8

Generators and relations for D8⋊3D11
 G = < a,b,c,d | a8=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 362 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C11, C2×C8, D8, SD16, Q16, C4○D4, D11, C22, C22, C4○D8, Dic11, Dic11, C44, D22, C2×C22, C11⋊C8, C88, Dic22, C4×D11, C2×Dic11, C11⋊D4, D4×C11, C8×D11, Dic44, D4.D11, C11×D8, D4⋊2D11, D8⋊3D11
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4○D8, D22, C22×D11, D4×D11, D8⋊3D11

Smallest permutation representation of D8⋊3D11
►On 176 points
Generators in S176
(1 164 32 153 21 175 43 142)(2 165 33 154 22 176 44 143)(3 155 23 144 12 166 34 133)(4 156 24 145 13 167 35 134)(5 157 25 146 14 168 36 135)(6 158 26 147 15 169 37 136)(7 159 27 148 16 170 38 137)(8 160 28 149 17 171 39 138)(9 161 29 150 18 172 40 139)(10 162 30 151 19 173 41 140)(11 163 31 152 20 174 42 141)(45 122 78 100 56 111 67 89)(46 123 79 101 57 112 68 90)(47 124 80 102 58 113 69 91)(48 125 81 103 59 114 70 92)(49 126 82 104 60 115 71 93)(50 127 83 105 61 116 72 94)(51 128 84 106 62 117 73 95)(52 129 85 107 63 118 74 96)(53 130 86 108 64 119 75 97)(54 131 87 109 65 120 76 98)(55 132 88 110 66 121 77 99)

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

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

49 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D11A···11E22A···22E22F···22O44A···44E88A···88J
order1222244444888811···1122···2222···2244···4488···88
size1144222111144442222222···22···28···84···44···4

49 irreducible representations

dim11111122222244
type++++++++++++-
imageC1C2C2C2C2C2D4D4D11C4○D8D22D22D4×D11D8⋊3D11
kernelD8⋊3D11C8×D11Dic44D4.D11C11×D8D4⋊2D11Dic11D22D8C11C8D4C2C1
# reps1112121154510510

Matrix representation of D8⋊3D11 ►in GL4(𝔽89) generated by

37000
277700
00880
00088
,
856000
22400
00880
00088
,
1000
0100
00331
001614
,
1000
128800
005547
007234
G:=sub<GL(4,GF(89))| [37,27,0,0,0,77,0,0,0,0,88,0,0,0,0,88],[85,22,0,0,60,4,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,33,16,0,0,1,14],[1,12,0,0,0,88,0,0,0,0,55,72,0,0,47,34] >;
 

D8⋊3D11 in GAP, Magma, Sage, TeX 

D_8\rtimes_3D_{11}
 
% in TeX
 
G:=Group("D8:3D11");
 
// GroupNames label
 
G:=SmallGroup(352,107);
 
// by ID
 
G=gap.SmallGroup(352,107);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,362,116,297,159,69,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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