metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.92D4, M4(2).3Dic5, C5⋊5(D4.C8), D4.(C5⋊2C8), C8○D4.2D5, (C5×D4).3C8, Q8.(C5⋊2C8), (C5×Q8).3C8, C20.33(C2×C8), (C2×C8).268D10, C8.34(C5⋊D4), C4○D4.1Dic5, C20.4C8⋊12C2, (C5×M4(2)).9C4, C10.34(C22⋊C8), (C2×C40).223C22, (C2×C10).15M4(2), C4.30(C23.D5), C20.143(C22⋊C4), C2.8(C20.55D4), C22.1(C4.Dic5), C4.3(C2×C5⋊2C8), (C5×C8○D4).2C2, (C5×C4○D4).5C4, (C2×C5⋊2C16)⋊14C2, (C2×C20).240(C2×C4), (C2×C4).41(C2×Dic5), SmallGroup(320,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.92D4
G = < a,b,c | a40=1, b4=a30, c2=a25, bab-1=cac-1=a9, cbc-1=a35b3 >
Smallest permutation representation of C40.92D4
►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 74 143 93 31 64 133 83 21 54 123 113 11 44 153 103)(2 43 144 102 32 73 134 92 22 63 124 82 12 53 154 112)(3 52 145 111 33 42 135 101 23 72 125 91 13 62 155 81)(4 61 146 120 34 51 136 110 24 41 126 100 14 71 156 90)(5 70 147 89 35 60 137 119 25 50 127 109 15 80 157 99)(6 79 148 98 36 69 138 88 26 59 128 118 16 49 158 108)(7 48 149 107 37 78 139 97 27 68 129 87 17 58 159 117)(8 57 150 116 38 47 140 106 28 77 130 96 18 67 160 86)(9 66 151 85 39 56 141 115 29 46 131 105 19 76 121 95)(10 75 152 94 40 65 142 84 30 55 132 114 20 45 122 104)
(1 88 26 113 11 98 36 83 21 108 6 93 31 118 16 103)(2 97 27 82 12 107 37 92 22 117 7 102 32 87 17 112)(3 106 28 91 13 116 38 101 23 86 8 111 33 96 18 81)(4 115 29 100 14 85 39 110 24 95 9 120 34 105 19 90)(5 84 30 109 15 94 40 119 25 104 10 89 35 114 20 99)(41 146 66 131 51 156 76 141 61 126 46 151 71 136 56 121)(42 155 67 140 52 125 77 150 62 135 47 160 72 145 57 130)(43 124 68 149 53 134 78 159 63 144 48 129 73 154 58 139)(44 133 69 158 54 143 79 128 64 153 49 138 74 123 59 148)(45 142 70 127 55 152 80 137 65 122 50 147 75 132 60 157)
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), (1,74,143,93,31,64,133,83,21,54,123,113,11,44,153,103)(2,43,144,102,32,73,134,92,22,63,124,82,12,53,154,112)(3,52,145,111,33,42,135,101,23,72,125,91,13,62,155,81)(4,61,146,120,34,51,136,110,24,41,126,100,14,71,156,90)(5,70,147,89,35,60,137,119,25,50,127,109,15,80,157,99)(6,79,148,98,36,69,138,88,26,59,128,118,16,49,158,108)(7,48,149,107,37,78,139,97,27,68,129,87,17,58,159,117)(8,57,150,116,38,47,140,106,28,77,130,96,18,67,160,86)(9,66,151,85,39,56,141,115,29,46,131,105,19,76,121,95)(10,75,152,94,40,65,142,84,30,55,132,114,20,45,122,104), (1,88,26,113,11,98,36,83,21,108,6,93,31,118,16,103)(2,97,27,82,12,107,37,92,22,117,7,102,32,87,17,112)(3,106,28,91,13,116,38,101,23,86,8,111,33,96,18,81)(4,115,29,100,14,85,39,110,24,95,9,120,34,105,19,90)(5,84,30,109,15,94,40,119,25,104,10,89,35,114,20,99)(41,146,66,131,51,156,76,141,61,126,46,151,71,136,56,121)(42,155,67,140,52,125,77,150,62,135,47,160,72,145,57,130)(43,124,68,149,53,134,78,159,63,144,48,129,73,154,58,139)(44,133,69,158,54,143,79,128,64,153,49,138,74,123,59,148)(45,142,70,127,55,152,80,137,65,122,50,147,75,132,60,157)>;
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), (1,74,143,93,31,64,133,83,21,54,123,113,11,44,153,103)(2,43,144,102,32,73,134,92,22,63,124,82,12,53,154,112)(3,52,145,111,33,42,135,101,23,72,125,91,13,62,155,81)(4,61,146,120,34,51,136,110,24,41,126,100,14,71,156,90)(5,70,147,89,35,60,137,119,25,50,127,109,15,80,157,99)(6,79,148,98,36,69,138,88,26,59,128,118,16,49,158,108)(7,48,149,107,37,78,139,97,27,68,129,87,17,58,159,117)(8,57,150,116,38,47,140,106,28,77,130,96,18,67,160,86)(9,66,151,85,39,56,141,115,29,46,131,105,19,76,121,95)(10,75,152,94,40,65,142,84,30,55,132,114,20,45,122,104), (1,88,26,113,11,98,36,83,21,108,6,93,31,118,16,103)(2,97,27,82,12,107,37,92,22,117,7,102,32,87,17,112)(3,106,28,91,13,116,38,101,23,86,8,111,33,96,18,81)(4,115,29,100,14,85,39,110,24,95,9,120,34,105,19,90)(5,84,30,109,15,94,40,119,25,104,10,89,35,114,20,99)(41,146,66,131,51,156,76,141,61,126,46,151,71,136,56,121)(42,155,67,140,52,125,77,150,62,135,47,160,72,145,57,130)(43,124,68,149,53,134,78,159,63,144,48,129,73,154,58,139)(44,133,69,158,54,143,79,128,64,153,49,138,74,123,59,148)(45,142,70,127,55,152,80,137,65,122,50,147,75,132,60,157) );
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)], [(1,74,143,93,31,64,133,83,21,54,123,113,11,44,153,103),(2,43,144,102,32,73,134,92,22,63,124,82,12,53,154,112),(3,52,145,111,33,42,135,101,23,72,125,91,13,62,155,81),(4,61,146,120,34,51,136,110,24,41,126,100,14,71,156,90),(5,70,147,89,35,60,137,119,25,50,127,109,15,80,157,99),(6,79,148,98,36,69,138,88,26,59,128,118,16,49,158,108),(7,48,149,107,37,78,139,97,27,68,129,87,17,58,159,117),(8,57,150,116,38,47,140,106,28,77,130,96,18,67,160,86),(9,66,151,85,39,56,141,115,29,46,131,105,19,76,121,95),(10,75,152,94,40,65,142,84,30,55,132,114,20,45,122,104)], [(1,88,26,113,11,98,36,83,21,108,6,93,31,118,16,103),(2,97,27,82,12,107,37,92,22,117,7,102,32,87,17,112),(3,106,28,91,13,116,38,101,23,86,8,111,33,96,18,81),(4,115,29,100,14,85,39,110,24,95,9,120,34,105,19,90),(5,84,30,109,15,94,40,119,25,104,10,89,35,114,20,99),(41,146,66,131,51,156,76,141,61,126,46,151,71,136,56,121),(42,155,67,140,52,125,77,150,62,135,47,160,72,145,57,130),(43,124,68,149,53,134,78,159,63,144,48,129,73,154,58,139),(44,133,69,158,54,143,79,128,64,153,49,138,74,123,59,148),(45,142,70,127,55,152,80,137,65,122,50,147,75,132,60,157)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | ··· | 10H | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D5 | M4(2) | D10 | Dic5 | Dic5 | C5⋊D4 | C5⋊2C8 | C5⋊2C8 | D4.C8 | C4.Dic5 | C40.92D4 |
| kernel | C40.92D4 | C2×C5⋊2C16 | C20.4C8 | C5×C8○D4 | C5×M4(2) | C5×C4○D4 | C5×D4 | C5×Q8 | C40 | C8○D4 | C2×C10 | C2×C8 | M4(2) | C4○D4 | C8 | D4 | Q8 | C5 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C40.92D4 ►in GL4(𝔽241) generated by
| 216 | 0 | 0 | 0 |
| 41 | 135 | 0 | 0 |
| 0 | 0 | 233 | 0 |
| 0 | 0 | 0 | 233 |
| 8 | 90 | 0 | 0 |
| 0 | 233 | 0 | 0 |
| 0 | 0 | 0 | 226 |
| 0 | 0 | 128 | 226 |
| 8 | 90 | 0 | 0 |
| 20 | 233 | 0 | 0 |
| 0 | 0 | 0 | 226 |
| 0 | 0 | 113 | 0 |
G:=sub<GL(4,GF(241))| [216,41,0,0,0,135,0,0,0,0,233,0,0,0,0,233],[8,0,0,0,90,233,0,0,0,0,0,128,0,0,226,226],[8,20,0,0,90,233,0,0,0,0,0,113,0,0,226,0] >;
C40.92D4 in GAP, Magma, Sage, TeX
C_{40}._{92}D_4 % in TeX
G:=Group("C40.92D4"); // GroupNames label
G:=SmallGroup(320,119);
// by ID
G=gap.SmallGroup(320,119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,1123,570,136,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=a^30,c^2=a^25,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^35*b^3>;
// generators/relations
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