direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×M6(2), C4.C80, C160⋊7C2, C32⋊3C10, C8.3C40, C22.C80, C16.2C20, C20.7C16, C40.15C8, C80.11C4, C80.30C22, C2.3(C2×C80), (C2×C4).5C40, (C2×C80).18C2, (C2×C16).8C10, (C2×C20).23C8, C4.13(C2×C40), (C2×C8).13C20, (C2×C10).3C16, C8.22(C2×C20), C16.7(C2×C10), (C2×C40).56C4, C20.87(C2×C8), C10.23(C2×C16), C40.132(C2×C4), SmallGroup(320,175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×M6(2)
G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b17 >
Smallest permutation representation of C5×M6(2)
►On 160 points
Generators in S160
(1 105 57 130 70)(2 106 58 131 71)(3 107 59 132 72)(4 108 60 133 73)(5 109 61 134 74)(6 110 62 135 75)(7 111 63 136 76)(8 112 64 137 77)(9 113 33 138 78)(10 114 34 139 79)(11 115 35 140 80)(12 116 36 141 81)(13 117 37 142 82)(14 118 38 143 83)(15 119 39 144 84)(16 120 40 145 85)(17 121 41 146 86)(18 122 42 147 87)(19 123 43 148 88)(20 124 44 149 89)(21 125 45 150 90)(22 126 46 151 91)(23 127 47 152 92)(24 128 48 153 93)(25 97 49 154 94)(26 98 50 155 95)(27 99 51 156 96)(28 100 52 157 65)(29 101 53 158 66)(30 102 54 159 67)(31 103 55 160 68)(32 104 56 129 69)
(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)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(34 50)(36 52)(38 54)(40 56)(42 58)(44 60)(46 62)(48 64)(65 81)(67 83)(69 85)(71 87)(73 89)(75 91)(77 93)(79 95)(98 114)(100 116)(102 118)(104 120)(106 122)(108 124)(110 126)(112 128)(129 145)(131 147)(133 149)(135 151)(137 153)(139 155)(141 157)(143 159)
G:=sub<Sym(160)| (1,105,57,130,70)(2,106,58,131,71)(3,107,59,132,72)(4,108,60,133,73)(5,109,61,134,74)(6,110,62,135,75)(7,111,63,136,76)(8,112,64,137,77)(9,113,33,138,78)(10,114,34,139,79)(11,115,35,140,80)(12,116,36,141,81)(13,117,37,142,82)(14,118,38,143,83)(15,119,39,144,84)(16,120,40,145,85)(17,121,41,146,86)(18,122,42,147,87)(19,123,43,148,88)(20,124,44,149,89)(21,125,45,150,90)(22,126,46,151,91)(23,127,47,152,92)(24,128,48,153,93)(25,97,49,154,94)(26,98,50,155,95)(27,99,51,156,96)(28,100,52,157,65)(29,101,53,158,66)(30,102,54,159,67)(31,103,55,160,68)(32,104,56,129,69), (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), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(34,50)(36,52)(38,54)(40,56)(42,58)(44,60)(46,62)(48,64)(65,81)(67,83)(69,85)(71,87)(73,89)(75,91)(77,93)(79,95)(98,114)(100,116)(102,118)(104,120)(106,122)(108,124)(110,126)(112,128)(129,145)(131,147)(133,149)(135,151)(137,153)(139,155)(141,157)(143,159)>;
G:=Group( (1,105,57,130,70)(2,106,58,131,71)(3,107,59,132,72)(4,108,60,133,73)(5,109,61,134,74)(6,110,62,135,75)(7,111,63,136,76)(8,112,64,137,77)(9,113,33,138,78)(10,114,34,139,79)(11,115,35,140,80)(12,116,36,141,81)(13,117,37,142,82)(14,118,38,143,83)(15,119,39,144,84)(16,120,40,145,85)(17,121,41,146,86)(18,122,42,147,87)(19,123,43,148,88)(20,124,44,149,89)(21,125,45,150,90)(22,126,46,151,91)(23,127,47,152,92)(24,128,48,153,93)(25,97,49,154,94)(26,98,50,155,95)(27,99,51,156,96)(28,100,52,157,65)(29,101,53,158,66)(30,102,54,159,67)(31,103,55,160,68)(32,104,56,129,69), (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), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(34,50)(36,52)(38,54)(40,56)(42,58)(44,60)(46,62)(48,64)(65,81)(67,83)(69,85)(71,87)(73,89)(75,91)(77,93)(79,95)(98,114)(100,116)(102,118)(104,120)(106,122)(108,124)(110,126)(112,128)(129,145)(131,147)(133,149)(135,151)(137,153)(139,155)(141,157)(143,159) );
G=PermutationGroup([[(1,105,57,130,70),(2,106,58,131,71),(3,107,59,132,72),(4,108,60,133,73),(5,109,61,134,74),(6,110,62,135,75),(7,111,63,136,76),(8,112,64,137,77),(9,113,33,138,78),(10,114,34,139,79),(11,115,35,140,80),(12,116,36,141,81),(13,117,37,142,82),(14,118,38,143,83),(15,119,39,144,84),(16,120,40,145,85),(17,121,41,146,86),(18,122,42,147,87),(19,123,43,148,88),(20,124,44,149,89),(21,125,45,150,90),(22,126,46,151,91),(23,127,47,152,92),(24,128,48,153,93),(25,97,49,154,94),(26,98,50,155,95),(27,99,51,156,96),(28,100,52,157,65),(29,101,53,158,66),(30,102,54,159,67),(31,103,55,160,68),(32,104,56,129,69)], [(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)], [(2,18),(4,20),(6,22),(8,24),(10,26),(12,28),(14,30),(16,32),(34,50),(36,52),(38,54),(40,56),(42,58),(44,60),(46,62),(48,64),(65,81),(67,83),(69,85),(71,87),(73,89),(75,91),(77,93),(79,95),(98,114),(100,116),(102,118),(104,120),(106,122),(108,124),(110,126),(112,128),(129,145),(131,147),(133,149),(135,151),(137,153),(139,155),(141,157),(143,159)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 32A | ··· | 32P | 40A | ··· | 40P | 40Q | ··· | 40X | 80A | ··· | 80AF | 80AG | ··· | 80AV | 160A | ··· | 160BL |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 | 80 | ··· | 80 | 160 | ··· | 160 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C16 | C16 | C20 | C20 | C40 | C40 | C80 | C80 | M6(2) | C5×M6(2) |
| kernel | C5×M6(2) | C160 | C2×C80 | C80 | C2×C40 | M6(2) | C40 | C2×C20 | C32 | C2×C16 | C20 | C2×C10 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 8 | 16 | 16 | 32 | 32 | 8 | 32 |
Matrix representation of C5×M6(2) ►in GL3(𝔽641) generated by
| 562 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 392 | 639 |
| 0 | 603 | 249 |
| 640 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 392 | 640 |
G:=sub<GL(3,GF(641))| [562,0,0,0,1,0,0,0,1],[1,0,0,0,392,603,0,639,249],[640,0,0,0,1,392,0,0,640] >;
C5×M6(2) in GAP, Magma, Sage, TeX
C_5\times M_6(2)
% in TeX
G:=Group("C5xM6(2)"); // GroupNames label
G:=SmallGroup(320,175);
// by ID
G=gap.SmallGroup(320,175);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,2269,80,102,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^17>;
// generators/relations
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