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G = C202Q8order 160 = 25·5

1st semidirect product of C20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202Q8, C4.4D20, C42Dic10, C20.27D4, C42.4D5, C51(C4⋊Q8), (C4×C20).2C2, C2.4(C2×D20), C10.1(C2×D4), C10.2(C2×Q8), (C2×C4).73D10, C4⋊Dic5.4C2, C2.4(C2×Dic10), (C2×C10).10C23, (C2×C20).85C22, (C2×Dic10).3C2, (C2×Dic5).1C22, C22.34(C22×D5), SmallGroup(160,90)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C202Q8
C1C5C10C2×C10C2×Dic5C2×Dic10 — C202Q8
C5C2×C10 — C202Q8
C1C22C42

Generators and relations for C202Q8
 G = < a,b,c | a20=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 192 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2 [×2], C4 [×6], C4 [×4], C22, C5, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×6], C2×C10, C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4⋊Dic5 [×4], C4×C20, C2×Dic10 [×2], C202Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, Dic10 [×4], D20 [×2], C22×D5, C2×Dic10 [×2], C2×D20, C202Q8

Smallest permutation representation of C202Q8
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 61 22 52)(2 62 23 53)(3 63 24 54)(4 64 25 55)(5 65 26 56)(6 66 27 57)(7 67 28 58)(8 68 29 59)(9 69 30 60)(10 70 31 41)(11 71 32 42)(12 72 33 43)(13 73 34 44)(14 74 35 45)(15 75 36 46)(16 76 37 47)(17 77 38 48)(18 78 39 49)(19 79 40 50)(20 80 21 51)(81 155 128 112)(82 156 129 113)(83 157 130 114)(84 158 131 115)(85 159 132 116)(86 160 133 117)(87 141 134 118)(88 142 135 119)(89 143 136 120)(90 144 137 101)(91 145 138 102)(92 146 139 103)(93 147 140 104)(94 148 121 105)(95 149 122 106)(96 150 123 107)(97 151 124 108)(98 152 125 109)(99 153 126 110)(100 154 127 111)
(1 82 22 129)(2 81 23 128)(3 100 24 127)(4 99 25 126)(5 98 26 125)(6 97 27 124)(7 96 28 123)(8 95 29 122)(9 94 30 121)(10 93 31 140)(11 92 32 139)(12 91 33 138)(13 90 34 137)(14 89 35 136)(15 88 36 135)(16 87 37 134)(17 86 38 133)(18 85 39 132)(19 84 40 131)(20 83 21 130)(41 147 70 104)(42 146 71 103)(43 145 72 102)(44 144 73 101)(45 143 74 120)(46 142 75 119)(47 141 76 118)(48 160 77 117)(49 159 78 116)(50 158 79 115)(51 157 80 114)(52 156 61 113)(53 155 62 112)(54 154 63 111)(55 153 64 110)(56 152 65 109)(57 151 66 108)(58 150 67 107)(59 149 68 106)(60 148 69 105)

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

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

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

C202Q8 is a maximal subgroup of
C4.Dic20  C20.47D8  C20.2D8  C42.F5  C409Q8  C20.14Q16  C408Q8  C85D20  C4.5D40  C204Q16  C8⋊Dic10  C42.14D10  C42.20D10  C8.D20  D4.9D20  C20⋊SD16  D203Q8  D204Q8  C4⋊Dic20  C20.7Q16  Dic104Q8  C20.50D8  C20.38SD16  D4.2D20  C20.48SD16  C20.23Q16  C207Q16  C42.62D10  C42.65D10  C42.68D10  C42.71D10  C20.16D8  C204SD16  C20.17D8  C20.SD16  C20.Q16  C203Q16  C20.11Q16  C42.274D10  C42.276D10  C42.89D10  C42.90D10  C42.92D10  C42.99D10  D4×Dic10  C42.106D10  D46Dic10  D2024D4  D46D20  C42.117D10  Q8×Dic10  Dic1010Q8  Q86Dic10  Q8×D20  D2010Q8  C42.135D10  C42.141D10  C42.144D10  C42.148D10  C42.155D10  C42.165D10  C42.238D10  D5×C4⋊Q8  C42.241D10  C60.48D4  C204Dic6  C608Q8
C202Q8 is a maximal quotient of
(C2×Dic5)⋊Q8  C10.(C4⋊Q8)  C409Q8  C408Q8  C40.13Q8  C8⋊Dic10  C207(C4⋊C4)  (C2×C20)⋊10Q8  C428Dic5  C60.48D4  C204Dic6  C608Q8

46 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J5A5B10A···10F20A···20X
order12224···444445510···1020···20
size11112···220202020222···22···2

46 irreducible representations

dim1111222222
type+++++-++-+
imageC1C2C2C2D4Q8D5D10Dic10D20
kernelC202Q8C4⋊Dic5C4×C20C2×Dic10C20C20C42C2×C4C4C4
# reps14122426168

Matrix representation of C202Q8 in GL4(𝔽41) generated by

40000
04000
002739
001611
,
1200
404000
0010
0001
,
311900
401000
00112
002230
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,27,16,0,0,39,11],[1,40,0,0,2,40,0,0,0,0,1,0,0,0,0,1],[31,40,0,0,19,10,0,0,0,0,11,22,0,0,2,30] >;

C202Q8 in GAP, Magma, Sage, TeX

C_{20}\rtimes_2Q_8
% in TeX

G:=Group("C20:2Q8");
// GroupNames label

G:=SmallGroup(160,90);
// by ID

G=gap.SmallGroup(160,90);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,218,50,4613]);
// Polycyclic

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

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