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G = C20⋊Q8order 160 = 25·5

The semidirect product of C20 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20⋊Q8, Dic51Q8, C41Dic10, Dic5.6D4, C52(C4⋊Q8), C4⋊C4.4D5, C2.4(Q8×D5), C2.11(D4×D5), C10.5(C2×Q8), (C2×C4).42D10, C10.22(C2×D4), C4⋊Dic5.11C2, (C2×C20).4C22, (C4×Dic5).1C2, C2.7(C2×Dic10), (C2×C10).29C23, (C2×Dic10).4C2, C10.D4.2C2, (C2×Dic5).8C22, C22.46(C22×D5), (C5×C4⋊C4).5C2, SmallGroup(160,109)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20⋊Q8
C1C5C10C2×C10C2×Dic5C4×Dic5 — C20⋊Q8
C5C2×C10 — C20⋊Q8
C1C22C4⋊C4

Generators and relations for C20⋊Q8
 G = < a,b,c | a20=b4=1, c2=b2, bab-1=a11, cac-1=a9, cbc-1=b-1 >

Subgroups: 192 in 68 conjugacy classes, 37 normal (19 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C20⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, Dic10, C22×D5, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8

Smallest permutation representation of C20⋊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 80 35 114)(2 71 36 105)(3 62 37 116)(4 73 38 107)(5 64 39 118)(6 75 40 109)(7 66 21 120)(8 77 22 111)(9 68 23 102)(10 79 24 113)(11 70 25 104)(12 61 26 115)(13 72 27 106)(14 63 28 117)(15 74 29 108)(16 65 30 119)(17 76 31 110)(18 67 32 101)(19 78 33 112)(20 69 34 103)(41 130 88 150)(42 121 89 141)(43 132 90 152)(44 123 91 143)(45 134 92 154)(46 125 93 145)(47 136 94 156)(48 127 95 147)(49 138 96 158)(50 129 97 149)(51 140 98 160)(52 131 99 151)(53 122 100 142)(54 133 81 153)(55 124 82 144)(56 135 83 155)(57 126 84 146)(58 137 85 157)(59 128 86 148)(60 139 87 159)
(1 86 35 59)(2 95 36 48)(3 84 37 57)(4 93 38 46)(5 82 39 55)(6 91 40 44)(7 100 21 53)(8 89 22 42)(9 98 23 51)(10 87 24 60)(11 96 25 49)(12 85 26 58)(13 94 27 47)(14 83 28 56)(15 92 29 45)(16 81 30 54)(17 90 31 43)(18 99 32 52)(19 88 33 41)(20 97 34 50)(61 137 115 157)(62 126 116 146)(63 135 117 155)(64 124 118 144)(65 133 119 153)(66 122 120 142)(67 131 101 151)(68 140 102 160)(69 129 103 149)(70 138 104 158)(71 127 105 147)(72 136 106 156)(73 125 107 145)(74 134 108 154)(75 123 109 143)(76 132 110 152)(77 121 111 141)(78 130 112 150)(79 139 113 159)(80 128 114 148)

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

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

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

C20⋊Q8 is a maximal subgroup of
Dic5.D8  Dic5.14D8  Dic5.5D8  D4⋊Dic10  C20⋊Q8⋊C2  Q8⋊Dic10  Dic5.3Q16  Dic5.9Q16  Q8⋊C4⋊D5  Dic10⋊Q8  C405Q8  C403Q8  D20⋊Q8  C402Q8  Dic102Q8  C404Q8  D202Q8  C10.12- 1+4  C10.2- 1+4  C10.102+ 1+4  C42.88D10  C42.90D10  C42.97D10  C42.98D10  D4×Dic10  D45Dic10  C42.228D10  C42.115D10  Q8×Dic10  Q85Dic10  C42.232D10  C42.133D10  C20⋊(C4○D4)  C10.362+ 1+4  C10.732- 1+4  C10.452+ 1+4  (Q8×Dic5)⋊C2  C10.502+ 1+4  C10.162- 1+4  Dic1021D4  C10.1182+ 1+4  C10.232- 1+4  C10.242- 1+4  C10.582+ 1+4  C10.792- 1+4  C10.812- 1+4  C10.822- 1+4  C10.842- 1+4  Dic107Q8  C42.236D10  C42.148D10  D207Q8  C42.154D10  C42.157D10  C42.159D10  C42.160D10  C42.164D10  C42.165D10  Dic109Q8  D5×C4⋊Q8  D208Q8  C42.174D10  Dic151Q8  Dic3⋊Dic10  C60⋊Q8  C20⋊Dic6  C4⋊Dic30
C20⋊Q8 is a maximal quotient of
(C2×C20)⋊Q8  C10.49(C4×D4)  (C2×Dic5)⋊Q8  C2.(C20⋊Q8)  C405Q8  C403Q8  C8.8Dic10  C402Q8  C404Q8  C8.6Dic10  C204(C4⋊C4)  C205(C4⋊C4)  (C2×C4)⋊Dic10  (C2×C20).54D4  C206(C4⋊C4)  Dic151Q8  Dic3⋊Dic10  C60⋊Q8  C20⋊Dic6  C4⋊Dic30

34 conjugacy classes

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

34 irreducible representations

dim11111122222244
type+++++++--++-+-
imageC1C2C2C2C2C2D4Q8Q8D5D10Dic10D4×D5Q8×D5
kernelC20⋊Q8C4×Dic5C10.D4C4⋊Dic5C5×C4⋊C4C2×Dic10Dic5Dic5C20C4⋊C4C2×C4C4C2C2
# reps11211222226822

Matrix representation of C20⋊Q8 in GL4(𝔽41) generated by

6200
23500
0071
00400
,
0100
40000
00119
003230
,
353900
39600
00922
00032
G:=sub<GL(4,GF(41))| [6,2,0,0,2,35,0,0,0,0,7,40,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,11,32,0,0,9,30],[35,39,0,0,39,6,0,0,0,0,9,0,0,0,22,32] >;

C20⋊Q8 in GAP, Magma, Sage, TeX

C_{20}\rtimes Q_8
% in TeX

G:=Group("C20:Q8");
// GroupNames label

G:=SmallGroup(160,109);
// by ID

G=gap.SmallGroup(160,109);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,218,188,50,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^2=b^2,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations

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