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G = Q8×C20order 160 = 25·5

Direct product of C20 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C20, C42.3C10, C4⋊C4.6C10, (C4×C20).9C2, C4.4(C2×C20), C2.2(Q8×C10), C20.52(C2×C4), (C2×Q8).5C10, C10.19(C2×Q8), C2.5(C22×C20), (Q8×C10).10C2, C10.40(C4○D4), C10.46(C22×C4), (C2×C10).74C23, (C2×C20).122C22, C22.8(C22×C10), C2.3(C5×C4○D4), (C5×C4⋊C4).13C2, (C2×C4).16(C2×C10), SmallGroup(160,180)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C20
C1C2C22C2×C10C2×C20C5×C4⋊C4 — Q8×C20
C1C2 — Q8×C20
C1C2×C20 — Q8×C20

Generators and relations for Q8×C20
 G = < a,b,c | a20=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2 [×3], C4 [×8], C4 [×3], C22, C5, C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×Q8, C20 [×8], C20 [×3], C2×C10, C4×Q8, C2×C20, C2×C20 [×6], C5×Q8 [×4], C4×C20 [×3], C5×C4⋊C4 [×3], Q8×C10, Q8×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], Q8 [×2], C23, C10 [×7], C22×C4, C2×Q8, C4○D4, C20 [×4], C2×C10 [×7], C4×Q8, C2×C20 [×6], C5×Q8 [×2], C22×C10, C22×C20, Q8×C10, C5×C4○D4, Q8×C20

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

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

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

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

Q8×C20 is a maximal subgroup of
C20.26Q16  C20.48SD16  C20.23Q16  Q8.3Dic10  C42.210D10  C42.56D10  Q8⋊D20  Q8.1D20  C42.59D10  C207Q16  Dic1010Q8  C42.122D10  Q85Dic10  Q86Dic10  C42.125D10  C42.126D10  Q85D20  Q86D20  C42.232D10  D2010Q8  C42.131D10  C42.132D10  C42.133D10  C42.134D10  C42.135D10  C42.136D10

100 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4P5A5B5C5D10A···10L20A···20P20Q···20BL
order122244444···4555510···1020···2020···20
size111111112···211111···11···12···2

100 irreducible representations

dim11111111112222
type++++-
imageC1C2C2C2C4C5C10C10C10C20Q8C4○D4C5×Q8C5×C4○D4
kernelQ8×C20C4×C20C5×C4⋊C4Q8×C10C5×Q8C4×Q8C42C4⋊C4C2×Q8Q8C20C10C4C2
# reps13318412124322288

Matrix representation of Q8×C20 in GL3(𝔽41) generated by

3200
020
002
,
4000
01240
02229
,
4000
01634
02525
G:=sub<GL(3,GF(41))| [32,0,0,0,2,0,0,0,2],[40,0,0,0,12,22,0,40,29],[40,0,0,0,16,25,0,34,25] >;

Q8×C20 in GAP, Magma, Sage, TeX

Q_8\times C_{20}
% in TeX

G:=Group("Q8xC20");
// GroupNames label

G:=SmallGroup(160,180);
// by ID

G=gap.SmallGroup(160,180);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,247,554]);
// Polycyclic

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

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