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G = C20.6Q8order 160 = 25·5

3rd non-split extension by C20 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.6Q8, C42.5D5, C4.6Dic10, (C4×C20).3C2, C10.3(C2×Q8), (C2×C4).74D10, C4⋊Dic5.5C2, C51(C42.C2), C10.2(C4○D4), C2.6(C4○D20), C2.5(C2×Dic10), (C2×C10).11C23, (C2×C20).72C22, C10.D4.1C2, (C2×Dic5).2C22, C22.35(C22×D5), SmallGroup(160,91)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.6Q8
C1C5C10C2×C10C2×Dic5C10.D4 — C20.6Q8
C5C2×C10 — C20.6Q8
C1C22C42

Generators and relations for C20.6Q8
 G = < a,b,c | a20=b4=1, c2=a10b2, ab=ba, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 144 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C42.C2, C2×Dic5, C2×C20, C2×C20, C10.D4, C4⋊Dic5, C4×C20, C20.6Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, D10, C42.C2, Dic10, C22×D5, C2×Dic10, C4○D20, C20.6Q8

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

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

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

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

C20.6Q8 is a maximal subgroup of
C42.2D10  C40.13Q8  C42.264D10  C8⋊Dic10  C42.14D10  C42.19D10  Dic10.3Q8  D20.3Q8  D4.3Dic10  Q8.3Dic10  C42.213D10  C42.215D10  C42.72D10  C42.76D10  C42.77D10  C42.274D10  C42.277D10  C42.89D10  C42.90D10  C42.94D10  C42.96D10  C42.100D10  D45Dic10  C42.105D10  C42.113D10  C42.118D10  C42.119D10  Dic1010Q8  Q85Dic10  D2010Q8  C42.132D10  C42.134D10  C42.140D10  C42.234D10  C42.145D10  C42.147D10  D5×C42.C2  C42.236D10  C42.157D10  C42.159D10  C42.161D10  C42.168D10  C42.174D10  C42.176D10  Dic3.3Dic10  C60.6Q8  C60.24Q8
C20.6Q8 is a maximal quotient of
C2.(C20⋊Q8)  (C2×C4).Dic10  C207(C4⋊C4)  C10.92(C4×D4)  C429Dic5  Dic3.3Dic10  C60.6Q8  C60.24Q8

46 conjugacy classes

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

46 irreducible representations

dim1111222222
type++++-++-
imageC1C2C2C2Q8D5C4○D4D10Dic10C4○D20
kernelC20.6Q8C10.D4C4⋊Dic5C4×C20C20C42C10C2×C4C4C2
# reps14212246816

Matrix representation of C20.6Q8 in GL4(𝔽41) generated by

253000
273900
00180
001516
,
23100
51800
00320
00299
,
391800
2200
001539
003126
G:=sub<GL(4,GF(41))| [25,27,0,0,30,39,0,0,0,0,18,15,0,0,0,16],[23,5,0,0,1,18,0,0,0,0,32,29,0,0,0,9],[39,2,0,0,18,2,0,0,0,0,15,31,0,0,39,26] >;

C20.6Q8 in GAP, Magma, Sage, TeX

C_{20}._6Q_8
% in TeX

G:=Group("C20.6Q8");
// GroupNames label

G:=SmallGroup(160,91);
// by ID

G=gap.SmallGroup(160,91);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,55,218,86,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^2=a^10*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations

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