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G = Dic5⋊Q8order 160 = 25·5

2nd semidirect product of Dic5 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic52Q8, C20.21D4, C53(C4⋊Q8), C2.8(Q8×D5), (C2×Q8).4D5, (C2×C4).55D10, C10.56(C2×D4), (Q8×C10).4C2, C10.15(C2×Q8), C4.10(C5⋊D4), (C4×Dic5).3C2, (C2×C10).56C23, (C2×C20).63C22, C10.D4.6C2, (C2×Dic10).10C2, C22.63(C22×D5), (C2×Dic5).20C22, C2.20(C2×C5⋊D4), SmallGroup(160,165)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic5⋊Q8
C1C5C10C2×C10C2×Dic5C4×Dic5 — Dic5⋊Q8
C5C2×C10 — Dic5⋊Q8
C1C22C2×Q8

Generators and relations for Dic5⋊Q8
 G = < a,b,c,d | a10=c4=1, b2=a5, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 176 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C5, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], C42, C4⋊C4 [×4], C2×Q8, C2×Q8, Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C4⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4×Dic5, C10.D4 [×4], C2×Dic10, Q8×C10, Dic5⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C5⋊D4 [×2], C22×D5, Q8×D5 [×2], C2×C5⋊D4, Dic5⋊Q8

Smallest permutation representation of Dic5⋊Q8
Regular action on 160 points
Generators in S160
(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)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)(21 143 26 148)(22 142 27 147)(23 141 28 146)(24 150 29 145)(25 149 30 144)(41 153 46 158)(42 152 47 157)(43 151 48 156)(44 160 49 155)(45 159 50 154)(51 99 56 94)(52 98 57 93)(53 97 58 92)(54 96 59 91)(55 95 60 100)(61 104 66 109)(62 103 67 108)(63 102 68 107)(64 101 69 106)(65 110 70 105)(71 119 76 114)(72 118 77 113)(73 117 78 112)(74 116 79 111)(75 115 80 120)(81 121 86 126)(82 130 87 125)(83 129 88 124)(84 128 89 123)(85 127 90 122)
(1 39 22 43)(2 40 23 44)(3 31 24 45)(4 32 25 46)(5 33 26 47)(6 34 27 48)(7 35 28 49)(8 36 29 50)(9 37 30 41)(10 38 21 42)(11 147 151 136)(12 148 152 137)(13 149 153 138)(14 150 154 139)(15 141 155 140)(16 142 156 131)(17 143 157 132)(18 144 158 133)(19 145 159 134)(20 146 160 135)(51 71 65 82)(52 72 66 83)(53 73 67 84)(54 74 68 85)(55 75 69 86)(56 76 70 87)(57 77 61 88)(58 78 62 89)(59 79 63 90)(60 80 64 81)(91 116 102 127)(92 117 103 128)(93 118 104 129)(94 119 105 130)(95 120 106 121)(96 111 107 122)(97 112 108 123)(98 113 109 124)(99 114 110 125)(100 115 101 126)
(1 59 22 63)(2 60 23 64)(3 51 24 65)(4 52 25 66)(5 53 26 67)(6 54 27 68)(7 55 28 69)(8 56 29 70)(9 57 30 61)(10 58 21 62)(11 122 151 111)(12 123 152 112)(13 124 153 113)(14 125 154 114)(15 126 155 115)(16 127 156 116)(17 128 157 117)(18 129 158 118)(19 130 159 119)(20 121 160 120)(31 82 45 71)(32 83 46 72)(33 84 47 73)(34 85 48 74)(35 86 49 75)(36 87 50 76)(37 88 41 77)(38 89 42 78)(39 90 43 79)(40 81 44 80)(91 142 102 131)(92 143 103 132)(93 144 104 133)(94 145 105 134)(95 146 106 135)(96 147 107 136)(97 148 108 137)(98 149 109 138)(99 150 110 139)(100 141 101 140)

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

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

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

Dic5⋊Q8 is a maximal subgroup of
(C2×Q8).D10  (Q8×C10).C4  Dic5.Q16  D20.4D4  Dic5.3Q16  Dic5⋊Q16  C408C4.C2  Dic5⋊SD16  Dic53SD16  C40.31D4  C4015D4  C40.26D4  Dic53Q16  C40.37D4  D20.40D4  2- 1+4.2D5  Dic1010Q8  C42.122D10  C42.232D10  C42.134D10  (Q8×Dic5)⋊C2  C10.502+ 1+4  C10.152- 1+4  D2022D4  Dic1021D4  C10.522+ 1+4  C10.222- 1+4  C10.582+ 1+4  C42.233D10  C42.137D10  C42.138D10  C42.139D10  C42.140D10  C42.141D10  Dic108Q8  Dic109Q8  D5×C4⋊Q8  C42.171D10  C42.174D10  C42.180D10  Q8×C5⋊D4  C10.442- 1+4  C10.1042- 1+4  C10.1052- 1+4  (C2×C20)⋊17D4  Dic15⋊Q8  Dic5⋊Dic6  Dic158Q8  Dic154Q8
Dic5⋊Q8 is a maximal quotient of
C10.96(C4×D4)  (C2×Dic5)⋊6Q8  C205(C4⋊C4)  (C2×C4)⋊Dic10  (C2×C20).287D4  (C2×C20).53D4  C42.215D10  C42.68D10  C20.17D8  C20.SD16  C42.76D10  C10.C22≀C2  (Q8×C10)⋊17C4  Dic15⋊Q8  Dic5⋊Dic6  Dic158Q8  Dic154Q8

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J5A5B10A···10F20A···20L
order122244444444445510···1020···20
size11112244101010102020222···24···4

34 irreducible representations

dim11111222224
type+++++-+++-
imageC1C2C2C2C2Q8D4D5D10C5⋊D4Q8×D5
kernelDic5⋊Q8C4×Dic5C10.D4C2×Dic10Q8×C10Dic5C20C2×Q8C2×C4C4C2
# reps11411422684

Matrix representation of Dic5⋊Q8 in GL4(𝔽41) generated by

344000
8100
00400
00040
,
123900
112900
0093
00032
,
173500
72400
003432
0017
,
1000
0100
003238
0009
G:=sub<GL(4,GF(41))| [34,8,0,0,40,1,0,0,0,0,40,0,0,0,0,40],[12,11,0,0,39,29,0,0,0,0,9,0,0,0,3,32],[17,7,0,0,35,24,0,0,0,0,34,1,0,0,32,7],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,38,9] >;

Dic5⋊Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes Q_8
% in TeX

G:=Group("Dic5:Q8");
// GroupNames label

G:=SmallGroup(160,165);
// by ID

G=gap.SmallGroup(160,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,362,116,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^4=1,b^2=a^5,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
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