metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20:3C8, C20.7Q8, C20.32D4, C4.16D20, C42.2D5, C4.7Dic10, C10.10M4(2), C5:3(C4:C8), C4:(C5:2C8), (C4xC20).4C2, C10.8(C4:C4), C10.16(C2xC8), (C2xC20).18C4, (C2xC4).89D10, (C2xC4).3Dic5, C2.1(C4:Dic5), C2.2(C4.Dic5), C22.8(C2xDic5), (C2xC20).103C22, C2.3(C2xC5:2C8), (C2xC5:2C8).8C2, (C2xC10).46(C2xC4), SmallGroup(160,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20:3C8
G = < a,b | a20=b8=1, bab-1=a-1 >
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, Q8, D5, C4:C4, C2xC8, M4(2), Dic5, D10, C4:C8, C5:2C8, Dic10, D20, C2xDic5, C2xC5:2C8, C4.Dic5, C4:Dic5, C20:3C8
Smallest permutation representation of C20:3C8
►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 22 136 151 81 63 117 42)(2 21 137 150 82 62 118 41)(3 40 138 149 83 61 119 60)(4 39 139 148 84 80 120 59)(5 38 140 147 85 79 101 58)(6 37 121 146 86 78 102 57)(7 36 122 145 87 77 103 56)(8 35 123 144 88 76 104 55)(9 34 124 143 89 75 105 54)(10 33 125 142 90 74 106 53)(11 32 126 141 91 73 107 52)(12 31 127 160 92 72 108 51)(13 30 128 159 93 71 109 50)(14 29 129 158 94 70 110 49)(15 28 130 157 95 69 111 48)(16 27 131 156 96 68 112 47)(17 26 132 155 97 67 113 46)(18 25 133 154 98 66 114 45)(19 24 134 153 99 65 115 44)(20 23 135 152 100 64 116 43)
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), (1,22,136,151,81,63,117,42)(2,21,137,150,82,62,118,41)(3,40,138,149,83,61,119,60)(4,39,139,148,84,80,120,59)(5,38,140,147,85,79,101,58)(6,37,121,146,86,78,102,57)(7,36,122,145,87,77,103,56)(8,35,123,144,88,76,104,55)(9,34,124,143,89,75,105,54)(10,33,125,142,90,74,106,53)(11,32,126,141,91,73,107,52)(12,31,127,160,92,72,108,51)(13,30,128,159,93,71,109,50)(14,29,129,158,94,70,110,49)(15,28,130,157,95,69,111,48)(16,27,131,156,96,68,112,47)(17,26,132,155,97,67,113,46)(18,25,133,154,98,66,114,45)(19,24,134,153,99,65,115,44)(20,23,135,152,100,64,116,43)>;
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), (1,22,136,151,81,63,117,42)(2,21,137,150,82,62,118,41)(3,40,138,149,83,61,119,60)(4,39,139,148,84,80,120,59)(5,38,140,147,85,79,101,58)(6,37,121,146,86,78,102,57)(7,36,122,145,87,77,103,56)(8,35,123,144,88,76,104,55)(9,34,124,143,89,75,105,54)(10,33,125,142,90,74,106,53)(11,32,126,141,91,73,107,52)(12,31,127,160,92,72,108,51)(13,30,128,159,93,71,109,50)(14,29,129,158,94,70,110,49)(15,28,130,157,95,69,111,48)(16,27,131,156,96,68,112,47)(17,26,132,155,97,67,113,46)(18,25,133,154,98,66,114,45)(19,24,134,153,99,65,115,44)(20,23,135,152,100,64,116,43) );
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)], [(1,22,136,151,81,63,117,42),(2,21,137,150,82,62,118,41),(3,40,138,149,83,61,119,60),(4,39,139,148,84,80,120,59),(5,38,140,147,85,79,101,58),(6,37,121,146,86,78,102,57),(7,36,122,145,87,77,103,56),(8,35,123,144,88,76,104,55),(9,34,124,143,89,75,105,54),(10,33,125,142,90,74,106,53),(11,32,126,141,91,73,107,52),(12,31,127,160,92,72,108,51),(13,30,128,159,93,71,109,50),(14,29,129,158,94,70,110,49),(15,28,130,157,95,69,111,48),(16,27,131,156,96,68,112,47),(17,26,132,155,97,67,113,46),(18,25,133,154,98,66,114,45),(19,24,134,153,99,65,115,44),(20,23,135,152,100,64,116,43)]])
C20:3C8 is a maximal subgroup of
Dic10:3C8 C40:6C8 C40:5C8 D20:3C8 C20.53D8 C20.39SD16 C4.Dic20 C20.47D8 C4.D40 C20.2D8 C20.57D8 C20.26Q16 C20.9D8 C20.5Q16 C20.10D8 C8xDic10 C40:11Q8 C8xD20 C8:6D20 C40:Q8 C8:9D20 D5xC4:C8 C42.200D10 C42.202D10 C20:5M4(2) C42.30D10 C42.31D10 C20:13M4(2) C42.6Dic5 C42.7Dic5 C42.43D10 C42.187D10 C20.50D8 C20.38SD16 D4.3Dic10 D4xC5:2C8 C42.47D10 C20:7M4(2) C20:7D8 D4.1D20 D4.2D20 C20.48SD16 C20.23Q16 Q8.3Dic10 Q8xC5:2C8 C42.210D10 Q8:D20 Q8.1D20 C20:7Q16 C42.61D10 D20.23D4 Dic10.4Q8 D20.4Q8 C20:2D8 Dic10:9D4 C20:5SD16 D20:5Q8 D20:6Q8 C20:Q16 Dic10:5Q8 Dic10:6Q8 C60.15Q8 C60:5C8
C20:3C8 is a maximal quotient of
C40:6C8 C40:5C8 C20:3C16 C40.7C8 (C2xC20):8C8 C60.15Q8 C60:5C8
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | D5 | M4(2) | Dic5 | D10 | C5:2C8 | Dic10 | D20 | C4.Dic5 |
| kernel | C20:3C8 | C2xC5:2C8 | C4xC20 | C2xC20 | C20 | C20 | C20 | C42 | C10 | C2xC4 | C2xC4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C20:3C8 ►in GL3(F41) generated by
| 40 | 0 | 0 |
| 0 | 28 | 2 |
| 0 | 39 | 16 |
| 14 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 39 | 27 |
G:=sub<GL(3,GF(41))| [40,0,0,0,28,39,0,2,16],[14,0,0,0,14,39,0,0,27] >;
C20:3C8 in GAP, Magma, Sage, TeX
C_{20}\rtimes_3C_8 % in TeX
G:=Group("C20:3C8"); // GroupNames label
G:=SmallGroup(160,11);
// by ID
G=gap.SmallGroup(160,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,4613]);
// Polycyclic
G:=Group<a,b|a^20=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export