metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊2Q8, C4.4D20, C4⋊2Dic10, C20.27D4, C42.4D5, C5⋊1(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
Generators and relations for C20⋊2Q8
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, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×Q8, Dic5, C20, C2×C10, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C4⋊Dic5, C4×C20, C2×Dic10, C20⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, Dic10, D20, C22×D5, C2×Dic10, C2×D20, C20⋊2Q8
►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 31 50 67)(2 32 51 68)(3 33 52 69)(4 34 53 70)(5 35 54 71)(6 36 55 72)(7 37 56 73)(8 38 57 74)(9 39 58 75)(10 40 59 76)(11 21 60 77)(12 22 41 78)(13 23 42 79)(14 24 43 80)(15 25 44 61)(16 26 45 62)(17 27 46 63)(18 28 47 64)(19 29 48 65)(20 30 49 66)(81 143 134 114)(82 144 135 115)(83 145 136 116)(84 146 137 117)(85 147 138 118)(86 148 139 119)(87 149 140 120)(88 150 121 101)(89 151 122 102)(90 152 123 103)(91 153 124 104)(92 154 125 105)(93 155 126 106)(94 156 127 107)(95 157 128 108)(96 158 129 109)(97 159 130 110)(98 160 131 111)(99 141 132 112)(100 142 133 113)
(1 101 50 150)(2 120 51 149)(3 119 52 148)(4 118 53 147)(5 117 54 146)(6 116 55 145)(7 115 56 144)(8 114 57 143)(9 113 58 142)(10 112 59 141)(11 111 60 160)(12 110 41 159)(13 109 42 158)(14 108 43 157)(15 107 44 156)(16 106 45 155)(17 105 46 154)(18 104 47 153)(19 103 48 152)(20 102 49 151)(21 131 77 98)(22 130 78 97)(23 129 79 96)(24 128 80 95)(25 127 61 94)(26 126 62 93)(27 125 63 92)(28 124 64 91)(29 123 65 90)(30 122 66 89)(31 121 67 88)(32 140 68 87)(33 139 69 86)(34 138 70 85)(35 137 71 84)(36 136 72 83)(37 135 73 82)(38 134 74 81)(39 133 75 100)(40 132 76 99)
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), (1,31,50,67)(2,32,51,68)(3,33,52,69)(4,34,53,70)(5,35,54,71)(6,36,55,72)(7,37,56,73)(8,38,57,74)(9,39,58,75)(10,40,59,76)(11,21,60,77)(12,22,41,78)(13,23,42,79)(14,24,43,80)(15,25,44,61)(16,26,45,62)(17,27,46,63)(18,28,47,64)(19,29,48,65)(20,30,49,66)(81,143,134,114)(82,144,135,115)(83,145,136,116)(84,146,137,117)(85,147,138,118)(86,148,139,119)(87,149,140,120)(88,150,121,101)(89,151,122,102)(90,152,123,103)(91,153,124,104)(92,154,125,105)(93,155,126,106)(94,156,127,107)(95,157,128,108)(96,158,129,109)(97,159,130,110)(98,160,131,111)(99,141,132,112)(100,142,133,113), (1,101,50,150)(2,120,51,149)(3,119,52,148)(4,118,53,147)(5,117,54,146)(6,116,55,145)(7,115,56,144)(8,114,57,143)(9,113,58,142)(10,112,59,141)(11,111,60,160)(12,110,41,159)(13,109,42,158)(14,108,43,157)(15,107,44,156)(16,106,45,155)(17,105,46,154)(18,104,47,153)(19,103,48,152)(20,102,49,151)(21,131,77,98)(22,130,78,97)(23,129,79,96)(24,128,80,95)(25,127,61,94)(26,126,62,93)(27,125,63,92)(28,124,64,91)(29,123,65,90)(30,122,66,89)(31,121,67,88)(32,140,68,87)(33,139,69,86)(34,138,70,85)(35,137,71,84)(36,136,72,83)(37,135,73,82)(38,134,74,81)(39,133,75,100)(40,132,76,99)>;
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), (1,31,50,67)(2,32,51,68)(3,33,52,69)(4,34,53,70)(5,35,54,71)(6,36,55,72)(7,37,56,73)(8,38,57,74)(9,39,58,75)(10,40,59,76)(11,21,60,77)(12,22,41,78)(13,23,42,79)(14,24,43,80)(15,25,44,61)(16,26,45,62)(17,27,46,63)(18,28,47,64)(19,29,48,65)(20,30,49,66)(81,143,134,114)(82,144,135,115)(83,145,136,116)(84,146,137,117)(85,147,138,118)(86,148,139,119)(87,149,140,120)(88,150,121,101)(89,151,122,102)(90,152,123,103)(91,153,124,104)(92,154,125,105)(93,155,126,106)(94,156,127,107)(95,157,128,108)(96,158,129,109)(97,159,130,110)(98,160,131,111)(99,141,132,112)(100,142,133,113), (1,101,50,150)(2,120,51,149)(3,119,52,148)(4,118,53,147)(5,117,54,146)(6,116,55,145)(7,115,56,144)(8,114,57,143)(9,113,58,142)(10,112,59,141)(11,111,60,160)(12,110,41,159)(13,109,42,158)(14,108,43,157)(15,107,44,156)(16,106,45,155)(17,105,46,154)(18,104,47,153)(19,103,48,152)(20,102,49,151)(21,131,77,98)(22,130,78,97)(23,129,79,96)(24,128,80,95)(25,127,61,94)(26,126,62,93)(27,125,63,92)(28,124,64,91)(29,123,65,90)(30,122,66,89)(31,121,67,88)(32,140,68,87)(33,139,69,86)(34,138,70,85)(35,137,71,84)(36,136,72,83)(37,135,73,82)(38,134,74,81)(39,133,75,100)(40,132,76,99) );
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)], [(1,31,50,67),(2,32,51,68),(3,33,52,69),(4,34,53,70),(5,35,54,71),(6,36,55,72),(7,37,56,73),(8,38,57,74),(9,39,58,75),(10,40,59,76),(11,21,60,77),(12,22,41,78),(13,23,42,79),(14,24,43,80),(15,25,44,61),(16,26,45,62),(17,27,46,63),(18,28,47,64),(19,29,48,65),(20,30,49,66),(81,143,134,114),(82,144,135,115),(83,145,136,116),(84,146,137,117),(85,147,138,118),(86,148,139,119),(87,149,140,120),(88,150,121,101),(89,151,122,102),(90,152,123,103),(91,153,124,104),(92,154,125,105),(93,155,126,106),(94,156,127,107),(95,157,128,108),(96,158,129,109),(97,159,130,110),(98,160,131,111),(99,141,132,112),(100,142,133,113)], [(1,101,50,150),(2,120,51,149),(3,119,52,148),(4,118,53,147),(5,117,54,146),(6,116,55,145),(7,115,56,144),(8,114,57,143),(9,113,58,142),(10,112,59,141),(11,111,60,160),(12,110,41,159),(13,109,42,158),(14,108,43,157),(15,107,44,156),(16,106,45,155),(17,105,46,154),(18,104,47,153),(19,103,48,152),(20,102,49,151),(21,131,77,98),(22,130,78,97),(23,129,79,96),(24,128,80,95),(25,127,61,94),(26,126,62,93),(27,125,63,92),(28,124,64,91),(29,123,65,90),(30,122,66,89),(31,121,67,88),(32,140,68,87),(33,139,69,86),(34,138,70,85),(35,137,71,84),(36,136,72,83),(37,135,73,82),(38,134,74,81),(39,133,75,100),(40,132,76,99)]])
C20⋊2Q8 is a maximal subgroup of
C4.Dic20 C20.47D8 C20.2D8 C42.F5 C40⋊9Q8 C20.14Q16 C40⋊8Q8 C8⋊5D20 C4.5D40 C20⋊4Q16 C8⋊Dic10 C42.14D10 C42.20D10 C8.D20 D4.9D20 C20⋊SD16 D20⋊3Q8 D20⋊4Q8 C4⋊Dic20 C20.7Q16 Dic10⋊4Q8 C20.50D8 C20.38SD16 D4.2D20 C20.48SD16 C20.23Q16 C20⋊7Q16 C42.62D10 C42.65D10 C42.68D10 C42.71D10 C20.16D8 C20⋊4SD16 C20.17D8 C20.SD16 C20.Q16 C20⋊3Q16 C20.11Q16 C42.274D10 C42.276D10 C42.89D10 C42.90D10 C42.92D10 C42.99D10 D4×Dic10 C42.106D10 D4⋊6Dic10 D20⋊24D4 D4⋊6D20 C42.117D10 Q8×Dic10 Dic10⋊10Q8 Q8⋊6Dic10 Q8×D20 D20⋊10Q8 C42.135D10 C42.141D10 C42.144D10 C42.148D10 C42.155D10 C42.165D10 C42.238D10 D5×C4⋊Q8 C42.241D10 C60.48D4 C20⋊4Dic6 C60⋊8Q8
C20⋊2Q8 is a maximal quotient of
(C2×Dic5)⋊Q8 C10.(C4⋊Q8) C40⋊9Q8 C40⋊8Q8 C40.13Q8 C8⋊Dic10 C20⋊7(C4⋊C4) (C2×C20)⋊10Q8 C42⋊8Dic5 C60.48D4 C20⋊4Dic6 C60⋊8Q8
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + |
| image | C1 | C2 | C2 | C2 | D4 | Q8 | D5 | D10 | Dic10 | D20 |
| kernel | C20⋊2Q8 | C4⋊Dic5 | C4×C20 | C2×Dic10 | C20 | C20 | C42 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 2 | 6 | 16 | 8 |
Matrix representation of C20⋊2Q8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 27 | 39 |
| 0 | 0 | 16 | 11 |
| 1 | 2 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 19 | 0 | 0 |
| 40 | 10 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 22 | 30 |
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] >;
C20⋊2Q8 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