metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.7Q8, C20.55D8, C8.7Dic10, C8.26(C4×D5), C5⋊2C16.1C4, C40.52(C2×C4), (C2×C20).93D4, C20.35(C4⋊C4), (C2×C10).2Q16, C5⋊3(C8.4Q8), C4.28(D4⋊D5), (C2×C8).248D10, C8.C4.1D5, C40.6C4.3C2, (C2×C40).36C22, C10.11(C2.D8), C4.4(C10.D4), C2.5(C10.D8), C22.1(C5⋊Q16), (C2×C5⋊2C16).1C2, (C5×C8.C4).1C2, (C2×C4).117(C5⋊D4), SmallGroup(320,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.7Q8
G = < a,b,c | a40=1, b4=a20, c2=a5b2, bab-1=a11, cac-1=a9, cbc-1=a5b3 >
Smallest permutation representation of C40.7Q8
►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 72 31 42 21 52 11 62)(2 43 32 53 22 63 12 73)(3 54 33 64 23 74 13 44)(4 65 34 75 24 45 14 55)(5 76 35 46 25 56 15 66)(6 47 36 57 26 67 16 77)(7 58 37 68 27 78 17 48)(8 69 38 79 28 49 18 59)(9 80 39 50 29 60 19 70)(10 51 40 61 30 71 20 41)(81 140 111 150 101 160 91 130)(82 151 112 121 102 131 92 141)(83 122 113 132 103 142 93 152)(84 133 114 143 104 153 94 123)(85 144 115 154 105 124 95 134)(86 155 116 125 106 135 96 145)(87 126 117 136 107 146 97 156)(88 137 118 147 108 157 98 127)(89 148 119 158 109 128 99 138)(90 159 120 129 110 139 100 149)
(1 102 36 97 31 92 26 87 21 82 16 117 11 112 6 107)(2 111 37 106 32 101 27 96 22 91 17 86 12 81 7 116)(3 120 38 115 33 110 28 105 23 100 18 95 13 90 8 85)(4 89 39 84 34 119 29 114 24 109 19 104 14 99 9 94)(5 98 40 93 35 88 30 83 25 118 20 113 15 108 10 103)(41 147 56 122 71 137 46 152 61 127 76 142 51 157 66 132)(42 156 57 131 72 146 47 121 62 136 77 151 52 126 67 141)(43 125 58 140 73 155 48 130 63 145 78 160 53 135 68 150)(44 134 59 149 74 124 49 139 64 154 79 129 54 144 69 159)(45 143 60 158 75 133 50 148 65 123 80 138 55 153 70 128)
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), (1,72,31,42,21,52,11,62)(2,43,32,53,22,63,12,73)(3,54,33,64,23,74,13,44)(4,65,34,75,24,45,14,55)(5,76,35,46,25,56,15,66)(6,47,36,57,26,67,16,77)(7,58,37,68,27,78,17,48)(8,69,38,79,28,49,18,59)(9,80,39,50,29,60,19,70)(10,51,40,61,30,71,20,41)(81,140,111,150,101,160,91,130)(82,151,112,121,102,131,92,141)(83,122,113,132,103,142,93,152)(84,133,114,143,104,153,94,123)(85,144,115,154,105,124,95,134)(86,155,116,125,106,135,96,145)(87,126,117,136,107,146,97,156)(88,137,118,147,108,157,98,127)(89,148,119,158,109,128,99,138)(90,159,120,129,110,139,100,149), (1,102,36,97,31,92,26,87,21,82,16,117,11,112,6,107)(2,111,37,106,32,101,27,96,22,91,17,86,12,81,7,116)(3,120,38,115,33,110,28,105,23,100,18,95,13,90,8,85)(4,89,39,84,34,119,29,114,24,109,19,104,14,99,9,94)(5,98,40,93,35,88,30,83,25,118,20,113,15,108,10,103)(41,147,56,122,71,137,46,152,61,127,76,142,51,157,66,132)(42,156,57,131,72,146,47,121,62,136,77,151,52,126,67,141)(43,125,58,140,73,155,48,130,63,145,78,160,53,135,68,150)(44,134,59,149,74,124,49,139,64,154,79,129,54,144,69,159)(45,143,60,158,75,133,50,148,65,123,80,138,55,153,70,128)>;
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), (1,72,31,42,21,52,11,62)(2,43,32,53,22,63,12,73)(3,54,33,64,23,74,13,44)(4,65,34,75,24,45,14,55)(5,76,35,46,25,56,15,66)(6,47,36,57,26,67,16,77)(7,58,37,68,27,78,17,48)(8,69,38,79,28,49,18,59)(9,80,39,50,29,60,19,70)(10,51,40,61,30,71,20,41)(81,140,111,150,101,160,91,130)(82,151,112,121,102,131,92,141)(83,122,113,132,103,142,93,152)(84,133,114,143,104,153,94,123)(85,144,115,154,105,124,95,134)(86,155,116,125,106,135,96,145)(87,126,117,136,107,146,97,156)(88,137,118,147,108,157,98,127)(89,148,119,158,109,128,99,138)(90,159,120,129,110,139,100,149), (1,102,36,97,31,92,26,87,21,82,16,117,11,112,6,107)(2,111,37,106,32,101,27,96,22,91,17,86,12,81,7,116)(3,120,38,115,33,110,28,105,23,100,18,95,13,90,8,85)(4,89,39,84,34,119,29,114,24,109,19,104,14,99,9,94)(5,98,40,93,35,88,30,83,25,118,20,113,15,108,10,103)(41,147,56,122,71,137,46,152,61,127,76,142,51,157,66,132)(42,156,57,131,72,146,47,121,62,136,77,151,52,126,67,141)(43,125,58,140,73,155,48,130,63,145,78,160,53,135,68,150)(44,134,59,149,74,124,49,139,64,154,79,129,54,144,69,159)(45,143,60,158,75,133,50,148,65,123,80,138,55,153,70,128) );
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)], [(1,72,31,42,21,52,11,62),(2,43,32,53,22,63,12,73),(3,54,33,64,23,74,13,44),(4,65,34,75,24,45,14,55),(5,76,35,46,25,56,15,66),(6,47,36,57,26,67,16,77),(7,58,37,68,27,78,17,48),(8,69,38,79,28,49,18,59),(9,80,39,50,29,60,19,70),(10,51,40,61,30,71,20,41),(81,140,111,150,101,160,91,130),(82,151,112,121,102,131,92,141),(83,122,113,132,103,142,93,152),(84,133,114,143,104,153,94,123),(85,144,115,154,105,124,95,134),(86,155,116,125,106,135,96,145),(87,126,117,136,107,146,97,156),(88,137,118,147,108,157,98,127),(89,148,119,158,109,128,99,138),(90,159,120,129,110,139,100,149)], [(1,102,36,97,31,92,26,87,21,82,16,117,11,112,6,107),(2,111,37,106,32,101,27,96,22,91,17,86,12,81,7,116),(3,120,38,115,33,110,28,105,23,100,18,95,13,90,8,85),(4,89,39,84,34,119,29,114,24,109,19,104,14,99,9,94),(5,98,40,93,35,88,30,83,25,118,20,113,15,108,10,103),(41,147,56,122,71,137,46,152,61,127,76,142,51,157,66,132),(42,156,57,131,72,146,47,121,62,136,77,151,52,126,67,141),(43,125,58,140,73,155,48,130,63,145,78,160,53,135,68,150),(44,134,59,149,74,124,49,139,64,154,79,129,54,144,69,159),(45,143,60,158,75,133,50,148,65,123,80,138,55,153,70,128)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | D8 | Q16 | D10 | Dic10 | C4×D5 | C5⋊D4 | C8.4Q8 | D4⋊D5 | C5⋊Q16 | C40.7Q8 |
| kernel | C40.7Q8 | C2×C5⋊2C16 | C40.6C4 | C5×C8.C4 | C5⋊2C16 | C40 | C2×C20 | C8.C4 | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 8 |
Matrix representation of C40.7Q8 ►in GL4(𝔽241) generated by
| 52 | 190 | 0 | 0 |
| 52 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 8 |
| 214 | 192 | 0 | 0 |
| 187 | 27 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
| 159 | 34 | 0 | 0 |
| 107 | 82 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 0 | 126 |
G:=sub<GL(4,GF(241))| [52,52,0,0,190,0,0,0,0,0,30,0,0,0,0,8],[214,187,0,0,192,27,0,0,0,0,0,8,0,0,8,0],[159,107,0,0,34,82,0,0,0,0,44,0,0,0,0,126] >;
C40.7Q8 in GAP, Magma, Sage, TeX
C_{40}._7Q_8 % in TeX
G:=Group("C40.7Q8"); // GroupNames label
G:=SmallGroup(320,51);
// by ID
G=gap.SmallGroup(320,51);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,141,36,184,346,192,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=a^20,c^2=a^5*b^2,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=a^5*b^3>;
// generators/relations
Export