Copied to
clipboard

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 [×3], C4 [×2], C4 [×8], C22, C5, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C4⋊Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×Dic10 [×2], C20⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, Dic10 [×2], 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)
(1 126 40 90)(2 137 21 81)(3 128 22 92)(4 139 23 83)(5 130 24 94)(6 121 25 85)(7 132 26 96)(8 123 27 87)(9 134 28 98)(10 125 29 89)(11 136 30 100)(12 127 31 91)(13 138 32 82)(14 129 33 93)(15 140 34 84)(16 131 35 95)(17 122 36 86)(18 133 37 97)(19 124 38 88)(20 135 39 99)(41 116 75 145)(42 107 76 156)(43 118 77 147)(44 109 78 158)(45 120 79 149)(46 111 80 160)(47 102 61 151)(48 113 62 142)(49 104 63 153)(50 115 64 144)(51 106 65 155)(52 117 66 146)(53 108 67 157)(54 119 68 148)(55 110 69 159)(56 101 70 150)(57 112 71 141)(58 103 72 152)(59 114 73 143)(60 105 74 154)
(1 116 40 145)(2 105 21 154)(3 114 22 143)(4 103 23 152)(5 112 24 141)(6 101 25 150)(7 110 26 159)(8 119 27 148)(9 108 28 157)(10 117 29 146)(11 106 30 155)(12 115 31 144)(13 104 32 153)(14 113 33 142)(15 102 34 151)(16 111 35 160)(17 120 36 149)(18 109 37 158)(19 118 38 147)(20 107 39 156)(41 90 75 126)(42 99 76 135)(43 88 77 124)(44 97 78 133)(45 86 79 122)(46 95 80 131)(47 84 61 140)(48 93 62 129)(49 82 63 138)(50 91 64 127)(51 100 65 136)(52 89 66 125)(53 98 67 134)(54 87 68 123)(55 96 69 132)(56 85 70 121)(57 94 71 130)(58 83 72 139)(59 92 73 128)(60 81 74 137)

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

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

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

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

׿
×
𝔽