metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.4D8, C40.81D4, C4.12D40, Dic20.4C4, M5(2).4D5, C8.5(C4×D5), C40.43(C2×C4), (C2×C8).47D10, (C2×C4).10D20, (C2×C20).100D4, C5⋊3(C8.17D4), C8.38(C5⋊D4), (C2×C10).9SD16, C40.6C4.6C2, (C2×C40).51C22, (C2×Dic20).6C2, (C5×M5(2)).5C2, C20.90(C22⋊C4), C22.7(C40⋊C2), C4.19(D10⋊C4), C10.33(D4⋊C4), C2.10(D20⋊5C4), SmallGroup(320,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.4D8
G = < a,b,c | a20=1, b8=a10, c2=a5, bab-1=a11, cac-1=a9, cbc-1=a5b7 >
2C2
20C4
20C4
2C10
10Q8
10Q8
20C2×C4
20Q8
20C8
4Dic5
4Dic5
2C16
5Q16
5Q16
10M4(2)
10Q16
10C2×Q8
2C80
Smallest permutation representation of C20.4D8
►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 71 36 51 16 76 31 56 11 61 26 41 6 66 21 46)(2 62 37 42 17 67 32 47 12 72 27 52 7 77 22 57)(3 73 38 53 18 78 33 58 13 63 28 43 8 68 23 48)(4 64 39 44 19 69 34 49 14 74 29 54 9 79 24 59)(5 75 40 55 20 80 35 60 15 65 30 45 10 70 25 50)(81 136 115 147 96 121 110 152 91 126 105 157 86 131 120 142)(82 127 116 158 97 132 111 143 92 137 106 148 87 122 101 153)(83 138 117 149 98 123 112 154 93 128 107 159 88 133 102 144)(84 129 118 160 99 134 113 145 94 139 108 150 89 124 103 155)(85 140 119 151 100 125 114 156 95 130 109 141 90 135 104 146)
(1 147 6 152 11 157 16 142)(2 156 7 141 12 146 17 151)(3 145 8 150 13 155 18 160)(4 154 9 159 14 144 19 149)(5 143 10 148 15 153 20 158)(21 121 26 126 31 131 36 136)(22 130 27 135 32 140 37 125)(23 139 28 124 33 129 38 134)(24 128 29 133 34 138 39 123)(25 137 30 122 35 127 40 132)(41 86 46 91 51 96 56 81)(42 95 47 100 52 85 57 90)(43 84 48 89 53 94 58 99)(44 93 49 98 54 83 59 88)(45 82 50 87 55 92 60 97)(61 120 66 105 71 110 76 115)(62 109 67 114 72 119 77 104)(63 118 68 103 73 108 78 113)(64 107 69 112 74 117 79 102)(65 116 70 101 75 106 80 111)
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), (1,71,36,51,16,76,31,56,11,61,26,41,6,66,21,46)(2,62,37,42,17,67,32,47,12,72,27,52,7,77,22,57)(3,73,38,53,18,78,33,58,13,63,28,43,8,68,23,48)(4,64,39,44,19,69,34,49,14,74,29,54,9,79,24,59)(5,75,40,55,20,80,35,60,15,65,30,45,10,70,25,50)(81,136,115,147,96,121,110,152,91,126,105,157,86,131,120,142)(82,127,116,158,97,132,111,143,92,137,106,148,87,122,101,153)(83,138,117,149,98,123,112,154,93,128,107,159,88,133,102,144)(84,129,118,160,99,134,113,145,94,139,108,150,89,124,103,155)(85,140,119,151,100,125,114,156,95,130,109,141,90,135,104,146), (1,147,6,152,11,157,16,142)(2,156,7,141,12,146,17,151)(3,145,8,150,13,155,18,160)(4,154,9,159,14,144,19,149)(5,143,10,148,15,153,20,158)(21,121,26,126,31,131,36,136)(22,130,27,135,32,140,37,125)(23,139,28,124,33,129,38,134)(24,128,29,133,34,138,39,123)(25,137,30,122,35,127,40,132)(41,86,46,91,51,96,56,81)(42,95,47,100,52,85,57,90)(43,84,48,89,53,94,58,99)(44,93,49,98,54,83,59,88)(45,82,50,87,55,92,60,97)(61,120,66,105,71,110,76,115)(62,109,67,114,72,119,77,104)(63,118,68,103,73,108,78,113)(64,107,69,112,74,117,79,102)(65,116,70,101,75,106,80,111)>;
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), (1,71,36,51,16,76,31,56,11,61,26,41,6,66,21,46)(2,62,37,42,17,67,32,47,12,72,27,52,7,77,22,57)(3,73,38,53,18,78,33,58,13,63,28,43,8,68,23,48)(4,64,39,44,19,69,34,49,14,74,29,54,9,79,24,59)(5,75,40,55,20,80,35,60,15,65,30,45,10,70,25,50)(81,136,115,147,96,121,110,152,91,126,105,157,86,131,120,142)(82,127,116,158,97,132,111,143,92,137,106,148,87,122,101,153)(83,138,117,149,98,123,112,154,93,128,107,159,88,133,102,144)(84,129,118,160,99,134,113,145,94,139,108,150,89,124,103,155)(85,140,119,151,100,125,114,156,95,130,109,141,90,135,104,146), (1,147,6,152,11,157,16,142)(2,156,7,141,12,146,17,151)(3,145,8,150,13,155,18,160)(4,154,9,159,14,144,19,149)(5,143,10,148,15,153,20,158)(21,121,26,126,31,131,36,136)(22,130,27,135,32,140,37,125)(23,139,28,124,33,129,38,134)(24,128,29,133,34,138,39,123)(25,137,30,122,35,127,40,132)(41,86,46,91,51,96,56,81)(42,95,47,100,52,85,57,90)(43,84,48,89,53,94,58,99)(44,93,49,98,54,83,59,88)(45,82,50,87,55,92,60,97)(61,120,66,105,71,110,76,115)(62,109,67,114,72,119,77,104)(63,118,68,103,73,108,78,113)(64,107,69,112,74,117,79,102)(65,116,70,101,75,106,80,111) );
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)], [(1,71,36,51,16,76,31,56,11,61,26,41,6,66,21,46),(2,62,37,42,17,67,32,47,12,72,27,52,7,77,22,57),(3,73,38,53,18,78,33,58,13,63,28,43,8,68,23,48),(4,64,39,44,19,69,34,49,14,74,29,54,9,79,24,59),(5,75,40,55,20,80,35,60,15,65,30,45,10,70,25,50),(81,136,115,147,96,121,110,152,91,126,105,157,86,131,120,142),(82,127,116,158,97,132,111,143,92,137,106,148,87,122,101,153),(83,138,117,149,98,123,112,154,93,128,107,159,88,133,102,144),(84,129,118,160,99,134,113,145,94,139,108,150,89,124,103,155),(85,140,119,151,100,125,114,156,95,130,109,141,90,135,104,146)], [(1,147,6,152,11,157,16,142),(2,156,7,141,12,146,17,151),(3,145,8,150,13,155,18,160),(4,154,9,159,14,144,19,149),(5,143,10,148,15,153,20,158),(21,121,26,126,31,131,36,136),(22,130,27,135,32,140,37,125),(23,139,28,124,33,129,38,134),(24,128,29,133,34,138,39,123),(25,137,30,122,35,127,40,132),(41,86,46,91,51,96,56,81),(42,95,47,100,52,85,57,90),(43,84,48,89,53,94,58,99),(44,93,49,98,54,83,59,88),(45,82,50,87,55,92,60,97),(61,120,66,105,71,110,76,115),(62,109,67,114,72,119,77,104),(63,118,68,103,73,108,78,113),(64,107,69,112,74,117,79,102),(65,116,70,101,75,106,80,111)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | C5⋊D4 | D20 | D40 | C40⋊C2 | C8.17D4 | C20.4D8 |
| kernel | C20.4D8 | C40.6C4 | C5×M5(2) | C2×Dic20 | Dic20 | C40 | C2×C20 | M5(2) | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of C20.4D8 ►in GL4(𝔽241) generated by
| 156 | 41 | 0 | 0 |
| 200 | 119 | 0 | 0 |
| 134 | 11 | 85 | 200 |
| 230 | 224 | 41 | 122 |
| 9 | 78 | 239 | 0 |
| 163 | 209 | 0 | 239 |
| 231 | 202 | 232 | 163 |
| 39 | 131 | 78 | 32 |
| 143 | 46 | 4 | 103 |
| 11 | 98 | 70 | 237 |
| 108 | 169 | 97 | 187 |
| 1 | 133 | 170 | 144 |
G:=sub<GL(4,GF(241))| [156,200,134,230,41,119,11,224,0,0,85,41,0,0,200,122],[9,163,231,39,78,209,202,131,239,0,232,78,0,239,163,32],[143,11,108,1,46,98,169,133,4,70,97,170,103,237,187,144] >;
C20.4D8 in GAP, Magma, Sage, TeX
C_{20}._4D_8 % in TeX
G:=Group("C20.4D8"); // GroupNames label
G:=SmallGroup(320,75);
// by ID
G=gap.SmallGroup(320,75);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,85,92,422,387,268,570,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^8=a^10,c^2=a^5,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=a^5*b^7>;
// generators/relations
Export