metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40.4C8, C5⋊2M6(2), C20.1C16, C5⋊C32⋊2C2, C4.(C5⋊C16), C8.2(C5⋊C8), (C2×C20).6C8, C22.(C5⋊C16), C8.40(C2×F5), (C2×C8).16F5, C5⋊2C16.6C4, C20.46(C2×C8), C10.9(C2×C16), C40.37(C2×C4), (C2×C40).17C4, (C2×C10).2C16, C5⋊2C16.16C22, C4.11(C2×C5⋊C8), C2.4(C2×C5⋊C16), (C2×C4).5(C5⋊C8), (C2×C5⋊2C16).12C2, SmallGroup(320,215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊M6(2)
G = < a,b,c | a5=b32=c2=1, bab-1=a3, ac=ca, cbc=b17 >
Smallest permutation representation of C5⋊M6(2)
►On 160 points
Generators in S160
(1 40 81 133 128)(2 134 41 97 82)(3 98 135 83 42)(4 84 99 43 136)(5 44 85 137 100)(6 138 45 101 86)(7 102 139 87 46)(8 88 103 47 140)(9 48 89 141 104)(10 142 49 105 90)(11 106 143 91 50)(12 92 107 51 144)(13 52 93 145 108)(14 146 53 109 94)(15 110 147 95 54)(16 96 111 55 148)(17 56 65 149 112)(18 150 57 113 66)(19 114 151 67 58)(20 68 115 59 152)(21 60 69 153 116)(22 154 61 117 70)(23 118 155 71 62)(24 72 119 63 156)(25 64 73 157 120)(26 158 33 121 74)(27 122 159 75 34)(28 76 123 35 160)(29 36 77 129 124)(30 130 37 125 78)(31 126 131 79 38)(32 80 127 39 132)
(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)(33 49)(35 51)(37 53)(39 55)(41 57)(43 59)(45 61)(47 63)(66 82)(68 84)(70 86)(72 88)(74 90)(76 92)(78 94)(80 96)(97 113)(99 115)(101 117)(103 119)(105 121)(107 123)(109 125)(111 127)(130 146)(132 148)(134 150)(136 152)(138 154)(140 156)(142 158)(144 160)
G:=sub<Sym(160)| (1,40,81,133,128)(2,134,41,97,82)(3,98,135,83,42)(4,84,99,43,136)(5,44,85,137,100)(6,138,45,101,86)(7,102,139,87,46)(8,88,103,47,140)(9,48,89,141,104)(10,142,49,105,90)(11,106,143,91,50)(12,92,107,51,144)(13,52,93,145,108)(14,146,53,109,94)(15,110,147,95,54)(16,96,111,55,148)(17,56,65,149,112)(18,150,57,113,66)(19,114,151,67,58)(20,68,115,59,152)(21,60,69,153,116)(22,154,61,117,70)(23,118,155,71,62)(24,72,119,63,156)(25,64,73,157,120)(26,158,33,121,74)(27,122,159,75,34)(28,76,123,35,160)(29,36,77,129,124)(30,130,37,125,78)(31,126,131,79,38)(32,80,127,39,132), (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)(33,49)(35,51)(37,53)(39,55)(41,57)(43,59)(45,61)(47,63)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)(97,113)(99,115)(101,117)(103,119)(105,121)(107,123)(109,125)(111,127)(130,146)(132,148)(134,150)(136,152)(138,154)(140,156)(142,158)(144,160)>;
G:=Group( (1,40,81,133,128)(2,134,41,97,82)(3,98,135,83,42)(4,84,99,43,136)(5,44,85,137,100)(6,138,45,101,86)(7,102,139,87,46)(8,88,103,47,140)(9,48,89,141,104)(10,142,49,105,90)(11,106,143,91,50)(12,92,107,51,144)(13,52,93,145,108)(14,146,53,109,94)(15,110,147,95,54)(16,96,111,55,148)(17,56,65,149,112)(18,150,57,113,66)(19,114,151,67,58)(20,68,115,59,152)(21,60,69,153,116)(22,154,61,117,70)(23,118,155,71,62)(24,72,119,63,156)(25,64,73,157,120)(26,158,33,121,74)(27,122,159,75,34)(28,76,123,35,160)(29,36,77,129,124)(30,130,37,125,78)(31,126,131,79,38)(32,80,127,39,132), (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)(33,49)(35,51)(37,53)(39,55)(41,57)(43,59)(45,61)(47,63)(66,82)(68,84)(70,86)(72,88)(74,90)(76,92)(78,94)(80,96)(97,113)(99,115)(101,117)(103,119)(105,121)(107,123)(109,125)(111,127)(130,146)(132,148)(134,150)(136,152)(138,154)(140,156)(142,158)(144,160) );
G=PermutationGroup([[(1,40,81,133,128),(2,134,41,97,82),(3,98,135,83,42),(4,84,99,43,136),(5,44,85,137,100),(6,138,45,101,86),(7,102,139,87,46),(8,88,103,47,140),(9,48,89,141,104),(10,142,49,105,90),(11,106,143,91,50),(12,92,107,51,144),(13,52,93,145,108),(14,146,53,109,94),(15,110,147,95,54),(16,96,111,55,148),(17,56,65,149,112),(18,150,57,113,66),(19,114,151,67,58),(20,68,115,59,152),(21,60,69,153,116),(22,154,61,117,70),(23,118,155,71,62),(24,72,119,63,156),(25,64,73,157,120),(26,158,33,121,74),(27,122,159,75,34),(28,76,123,35,160),(29,36,77,129,124),(30,130,37,125,78),(31,126,131,79,38),(32,80,127,39,132)], [(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),(33,49),(35,51),(37,53),(39,55),(41,57),(43,59),(45,61),(47,63),(66,82),(68,84),(70,86),(72,88),(74,90),(76,92),(78,94),(80,96),(97,113),(99,115),(101,117),(103,119),(105,121),(107,123),(109,125),(111,127),(130,146),(132,148),(134,150),(136,152),(138,154),(140,156),(142,158),(144,160)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 20A | 20B | 20C | 20D | 32A | ··· | 32P | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | M6(2) | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C5⋊C16 | C5⋊C16 | C5⋊M6(2) |
| kernel | C5⋊M6(2) | C5⋊C32 | C2×C5⋊2C16 | C5⋊2C16 | C2×C40 | C40 | C2×C20 | C20 | C2×C10 | C5 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 8 |
Matrix representation of C5⋊M6(2) ►in GL6(𝔽641)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 362 | 1 | 0 | 0 |
| 0 | 0 | 640 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 640 | 1 |
| 0 | 0 | 2 | 0 | 361 | 279 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 100 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 467 | 83 | 24 | 257 |
| 0 | 0 | 342 | 380 | 197 | 379 |
| 0 | 0 | 531 | 153 | 560 | 342 |
| 0 | 0 | 379 | 450 | 469 | 516 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 640 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(641))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,362,640,0,2,0,0,1,0,0,0,0,0,0,0,640,361,0,0,0,0,1,279],[0,100,0,0,0,0,1,0,0,0,0,0,0,0,467,342,531,379,0,0,83,380,153,450,0,0,24,197,560,469,0,0,257,379,342,516],[1,0,0,0,0,0,0,640,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C5⋊M6(2) in GAP, Magma, Sage, TeX
C_5\rtimes M_6(2)
% in TeX
G:=Group("C5:M6(2)"); // GroupNames label
G:=SmallGroup(320,215);
// by ID
G=gap.SmallGroup(320,215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,58,80,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=b^17>;
// generators/relations
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