metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊Q8, Dic5⋊1Q8, C4⋊1Dic10, Dic5.6D4, C5⋊2(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
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, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C20⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, Dic10, C22×D5, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8
►Regular action on 160 points
(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 80 35 114)(2 71 36 105)(3 62 37 116)(4 73 38 107)(5 64 39 118)(6 75 40 109)(7 66 21 120)(8 77 22 111)(9 68 23 102)(10 79 24 113)(11 70 25 104)(12 61 26 115)(13 72 27 106)(14 63 28 117)(15 74 29 108)(16 65 30 119)(17 76 31 110)(18 67 32 101)(19 78 33 112)(20 69 34 103)(41 130 88 150)(42 121 89 141)(43 132 90 152)(44 123 91 143)(45 134 92 154)(46 125 93 145)(47 136 94 156)(48 127 95 147)(49 138 96 158)(50 129 97 149)(51 140 98 160)(52 131 99 151)(53 122 100 142)(54 133 81 153)(55 124 82 144)(56 135 83 155)(57 126 84 146)(58 137 85 157)(59 128 86 148)(60 139 87 159)
(1 86 35 59)(2 95 36 48)(3 84 37 57)(4 93 38 46)(5 82 39 55)(6 91 40 44)(7 100 21 53)(8 89 22 42)(9 98 23 51)(10 87 24 60)(11 96 25 49)(12 85 26 58)(13 94 27 47)(14 83 28 56)(15 92 29 45)(16 81 30 54)(17 90 31 43)(18 99 32 52)(19 88 33 41)(20 97 34 50)(61 137 115 157)(62 126 116 146)(63 135 117 155)(64 124 118 144)(65 133 119 153)(66 122 120 142)(67 131 101 151)(68 140 102 160)(69 129 103 149)(70 138 104 158)(71 127 105 147)(72 136 106 156)(73 125 107 145)(74 134 108 154)(75 123 109 143)(76 132 110 152)(77 121 111 141)(78 130 112 150)(79 139 113 159)(80 128 114 148)
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), (1,80,35,114)(2,71,36,105)(3,62,37,116)(4,73,38,107)(5,64,39,118)(6,75,40,109)(7,66,21,120)(8,77,22,111)(9,68,23,102)(10,79,24,113)(11,70,25,104)(12,61,26,115)(13,72,27,106)(14,63,28,117)(15,74,29,108)(16,65,30,119)(17,76,31,110)(18,67,32,101)(19,78,33,112)(20,69,34,103)(41,130,88,150)(42,121,89,141)(43,132,90,152)(44,123,91,143)(45,134,92,154)(46,125,93,145)(47,136,94,156)(48,127,95,147)(49,138,96,158)(50,129,97,149)(51,140,98,160)(52,131,99,151)(53,122,100,142)(54,133,81,153)(55,124,82,144)(56,135,83,155)(57,126,84,146)(58,137,85,157)(59,128,86,148)(60,139,87,159), (1,86,35,59)(2,95,36,48)(3,84,37,57)(4,93,38,46)(5,82,39,55)(6,91,40,44)(7,100,21,53)(8,89,22,42)(9,98,23,51)(10,87,24,60)(11,96,25,49)(12,85,26,58)(13,94,27,47)(14,83,28,56)(15,92,29,45)(16,81,30,54)(17,90,31,43)(18,99,32,52)(19,88,33,41)(20,97,34,50)(61,137,115,157)(62,126,116,146)(63,135,117,155)(64,124,118,144)(65,133,119,153)(66,122,120,142)(67,131,101,151)(68,140,102,160)(69,129,103,149)(70,138,104,158)(71,127,105,147)(72,136,106,156)(73,125,107,145)(74,134,108,154)(75,123,109,143)(76,132,110,152)(77,121,111,141)(78,130,112,150)(79,139,113,159)(80,128,114,148)>;
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), (1,80,35,114)(2,71,36,105)(3,62,37,116)(4,73,38,107)(5,64,39,118)(6,75,40,109)(7,66,21,120)(8,77,22,111)(9,68,23,102)(10,79,24,113)(11,70,25,104)(12,61,26,115)(13,72,27,106)(14,63,28,117)(15,74,29,108)(16,65,30,119)(17,76,31,110)(18,67,32,101)(19,78,33,112)(20,69,34,103)(41,130,88,150)(42,121,89,141)(43,132,90,152)(44,123,91,143)(45,134,92,154)(46,125,93,145)(47,136,94,156)(48,127,95,147)(49,138,96,158)(50,129,97,149)(51,140,98,160)(52,131,99,151)(53,122,100,142)(54,133,81,153)(55,124,82,144)(56,135,83,155)(57,126,84,146)(58,137,85,157)(59,128,86,148)(60,139,87,159), (1,86,35,59)(2,95,36,48)(3,84,37,57)(4,93,38,46)(5,82,39,55)(6,91,40,44)(7,100,21,53)(8,89,22,42)(9,98,23,51)(10,87,24,60)(11,96,25,49)(12,85,26,58)(13,94,27,47)(14,83,28,56)(15,92,29,45)(16,81,30,54)(17,90,31,43)(18,99,32,52)(19,88,33,41)(20,97,34,50)(61,137,115,157)(62,126,116,146)(63,135,117,155)(64,124,118,144)(65,133,119,153)(66,122,120,142)(67,131,101,151)(68,140,102,160)(69,129,103,149)(70,138,104,158)(71,127,105,147)(72,136,106,156)(73,125,107,145)(74,134,108,154)(75,123,109,143)(76,132,110,152)(77,121,111,141)(78,130,112,150)(79,139,113,159)(80,128,114,148) );
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)], [(1,80,35,114),(2,71,36,105),(3,62,37,116),(4,73,38,107),(5,64,39,118),(6,75,40,109),(7,66,21,120),(8,77,22,111),(9,68,23,102),(10,79,24,113),(11,70,25,104),(12,61,26,115),(13,72,27,106),(14,63,28,117),(15,74,29,108),(16,65,30,119),(17,76,31,110),(18,67,32,101),(19,78,33,112),(20,69,34,103),(41,130,88,150),(42,121,89,141),(43,132,90,152),(44,123,91,143),(45,134,92,154),(46,125,93,145),(47,136,94,156),(48,127,95,147),(49,138,96,158),(50,129,97,149),(51,140,98,160),(52,131,99,151),(53,122,100,142),(54,133,81,153),(55,124,82,144),(56,135,83,155),(57,126,84,146),(58,137,85,157),(59,128,86,148),(60,139,87,159)], [(1,86,35,59),(2,95,36,48),(3,84,37,57),(4,93,38,46),(5,82,39,55),(6,91,40,44),(7,100,21,53),(8,89,22,42),(9,98,23,51),(10,87,24,60),(11,96,25,49),(12,85,26,58),(13,94,27,47),(14,83,28,56),(15,92,29,45),(16,81,30,54),(17,90,31,43),(18,99,32,52),(19,88,33,41),(20,97,34,50),(61,137,115,157),(62,126,116,146),(63,135,117,155),(64,124,118,144),(65,133,119,153),(66,122,120,142),(67,131,101,151),(68,140,102,160),(69,129,103,149),(70,138,104,158),(71,127,105,147),(72,136,106,156),(73,125,107,145),(74,134,108,154),(75,123,109,143),(76,132,110,152),(77,121,111,141),(78,130,112,150),(79,139,113,159),(80,128,114,148)]])
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 C40⋊5Q8 C40⋊3Q8 D20⋊Q8 C40⋊2Q8 Dic10⋊2Q8 C40⋊4Q8 D20⋊2Q8 C10.12- 1+4 C10.2- 1+4 C10.102+ 1+4 C42.88D10 C42.90D10 C42.97D10 C42.98D10 D4×Dic10 D4⋊5Dic10 C42.228D10 C42.115D10 Q8×Dic10 Q8⋊5Dic10 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 Dic10⋊21D4 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 Dic10⋊7Q8 C42.236D10 C42.148D10 D20⋊7Q8 C42.154D10 C42.157D10 C42.159D10 C42.160D10 C42.164D10 C42.165D10 Dic10⋊9Q8 D5×C4⋊Q8 D20⋊8Q8 C42.174D10 Dic15⋊1Q8 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) C40⋊5Q8 C40⋊3Q8 C8.8Dic10 C40⋊2Q8 C40⋊4Q8 C8.6Dic10 C20⋊4(C4⋊C4) C20⋊5(C4⋊C4) (C2×C4)⋊Dic10 (C2×C20).54D4 C20⋊6(C4⋊C4) Dic15⋊1Q8 Dic3⋊Dic10 C60⋊Q8 C20⋊Dic6 C4⋊Dic30
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D5 | D10 | Dic10 | D4×D5 | Q8×D5 |
| kernel | C20⋊Q8 | C4×Dic5 | C10.D4 | C4⋊Dic5 | C5×C4⋊C4 | C2×Dic10 | Dic5 | Dic5 | C20 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of C20⋊Q8 ►in GL4(𝔽41) generated by
| 6 | 2 | 0 | 0 |
| 2 | 35 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 32 | 30 |
| 35 | 39 | 0 | 0 |
| 39 | 6 | 0 | 0 |
| 0 | 0 | 9 | 22 |
| 0 | 0 | 0 | 32 |
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