metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5⋊C32, C10.C16, C8.4F5, C20.2C8, C40.3C4, C2.(C5⋊C16), C4.2(C5⋊C8), C5⋊2C16.2C2, SmallGroup(160,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5⋊C32 |
Generators and relations for C5⋊C32
G = < a,b | a5=b32=1, bab-1=a3 >
Smallest permutation representation of C5⋊C32
►Regular action on 160 points
Generators in S160
(1 104 59 75 129)(2 76 105 130 60)(3 131 77 61 106)(4 62 132 107 78)(5 108 63 79 133)(6 80 109 134 64)(7 135 81 33 110)(8 34 136 111 82)(9 112 35 83 137)(10 84 113 138 36)(11 139 85 37 114)(12 38 140 115 86)(13 116 39 87 141)(14 88 117 142 40)(15 143 89 41 118)(16 42 144 119 90)(17 120 43 91 145)(18 92 121 146 44)(19 147 93 45 122)(20 46 148 123 94)(21 124 47 95 149)(22 96 125 150 48)(23 151 65 49 126)(24 50 152 127 66)(25 128 51 67 153)(26 68 97 154 52)(27 155 69 53 98)(28 54 156 99 70)(29 100 55 71 157)(30 72 101 158 56)(31 159 73 57 102)(32 58 160 103 74)
(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)
G:=sub<Sym(160)| (1,104,59,75,129)(2,76,105,130,60)(3,131,77,61,106)(4,62,132,107,78)(5,108,63,79,133)(6,80,109,134,64)(7,135,81,33,110)(8,34,136,111,82)(9,112,35,83,137)(10,84,113,138,36)(11,139,85,37,114)(12,38,140,115,86)(13,116,39,87,141)(14,88,117,142,40)(15,143,89,41,118)(16,42,144,119,90)(17,120,43,91,145)(18,92,121,146,44)(19,147,93,45,122)(20,46,148,123,94)(21,124,47,95,149)(22,96,125,150,48)(23,151,65,49,126)(24,50,152,127,66)(25,128,51,67,153)(26,68,97,154,52)(27,155,69,53,98)(28,54,156,99,70)(29,100,55,71,157)(30,72,101,158,56)(31,159,73,57,102)(32,58,160,103,74), (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)>;
G:=Group( (1,104,59,75,129)(2,76,105,130,60)(3,131,77,61,106)(4,62,132,107,78)(5,108,63,79,133)(6,80,109,134,64)(7,135,81,33,110)(8,34,136,111,82)(9,112,35,83,137)(10,84,113,138,36)(11,139,85,37,114)(12,38,140,115,86)(13,116,39,87,141)(14,88,117,142,40)(15,143,89,41,118)(16,42,144,119,90)(17,120,43,91,145)(18,92,121,146,44)(19,147,93,45,122)(20,46,148,123,94)(21,124,47,95,149)(22,96,125,150,48)(23,151,65,49,126)(24,50,152,127,66)(25,128,51,67,153)(26,68,97,154,52)(27,155,69,53,98)(28,54,156,99,70)(29,100,55,71,157)(30,72,101,158,56)(31,159,73,57,102)(32,58,160,103,74), (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) );
G=PermutationGroup([[(1,104,59,75,129),(2,76,105,130,60),(3,131,77,61,106),(4,62,132,107,78),(5,108,63,79,133),(6,80,109,134,64),(7,135,81,33,110),(8,34,136,111,82),(9,112,35,83,137),(10,84,113,138,36),(11,139,85,37,114),(12,38,140,115,86),(13,116,39,87,141),(14,88,117,142,40),(15,143,89,41,118),(16,42,144,119,90),(17,120,43,91,145),(18,92,121,146,44),(19,147,93,45,122),(20,46,148,123,94),(21,124,47,95,149),(22,96,125,150,48),(23,151,65,49,126),(24,50,152,127,66),(25,128,51,67,153),(26,68,97,154,52),(27,155,69,53,98),(28,54,156,99,70),(29,100,55,71,157),(30,72,101,158,56),(31,159,73,57,102),(32,58,160,103,74)], [(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)]])
C5⋊C32 is a maximal subgroup of
D5⋊C32 C80.C4 C5⋊M6(2) C15⋊C32
C5⋊C32 is a maximal quotient of C5⋊C64 C15⋊C32
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | 16A | ··· | 16H | 20A | 20B | 32A | ··· | 32P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 16 | ··· | 16 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 5 | ··· | 5 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | F5 | C5⋊C8 | C5⋊C16 | C5⋊C32 |
| kernel | C5⋊C32 | C5⋊2C16 | C40 | C20 | C10 | C5 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 1 | 2 | 4 |
Matrix representation of C5⋊C32 ►in GL4(𝔽641) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 640 | 640 | 640 | 640 |
| 468 | 335 | 19 | 54 |
| 325 | 360 | 306 | 133 |
| 587 | 414 | 281 | 606 |
| 508 | 192 | 227 | 173 |
G:=sub<GL(4,GF(641))| [0,0,0,640,1,0,0,640,0,1,0,640,0,0,1,640],[468,325,587,508,335,360,414,192,19,306,281,227,54,133,606,173] >;
C5⋊C32 in GAP, Magma, Sage, TeX
C_5\rtimes C_{32} % in TeX
G:=Group("C5:C32"); // GroupNames label
G:=SmallGroup(160,3);
// by ID
G=gap.SmallGroup(160,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,12,31,50,69,2309,2315]);
// Polycyclic
G:=Group<a,b|a^5=b^32=1,b*a*b^-1=a^3>;
// generators/relations
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