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, C5⋊1(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
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
►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 D4⋊5Dic10 C42.105D10 C42.113D10 C42.118D10 C42.119D10 Dic10⋊10Q8 Q8⋊5Dic10 D20⋊10Q8 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 C20⋊7(C4⋊C4) C10.92(C4×D4) C42⋊9Dic5 Dic3.3Dic10 C60.6Q8 C60.24Q8
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 | Q8 | D5 | C4○D4 | D10 | Dic10 | C4○D20 |
| kernel | C20.6Q8 | C10.D4 | C4⋊Dic5 | C4×C20 | C20 | C42 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 2 | 4 | 6 | 8 | 16 |
Matrix representation of C20.6Q8 ►in GL4(𝔽41) generated by
| 25 | 30 | 0 | 0 |
| 27 | 39 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 15 | 16 |
| 23 | 1 | 0 | 0 |
| 5 | 18 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 29 | 9 |
| 39 | 18 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 0 | 0 | 15 | 39 |
| 0 | 0 | 31 | 26 |
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