direct product, p-group, abelian, monomial
Aliases: C4×C32, SmallGroup(128,128)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C4×C32 |
| C1 — C4×C32 |
| C1 — C4×C32 |
Generators and relations for C4×C32
G = < a,b | a4=b32=1, ab=ba >
Smallest permutation representation of C4×C32
►Regular action on 128 points
Generators in S128
(1 52 92 108)(2 53 93 109)(3 54 94 110)(4 55 95 111)(5 56 96 112)(6 57 65 113)(7 58 66 114)(8 59 67 115)(9 60 68 116)(10 61 69 117)(11 62 70 118)(12 63 71 119)(13 64 72 120)(14 33 73 121)(15 34 74 122)(16 35 75 123)(17 36 76 124)(18 37 77 125)(19 38 78 126)(20 39 79 127)(21 40 80 128)(22 41 81 97)(23 42 82 98)(24 43 83 99)(25 44 84 100)(26 45 85 101)(27 46 86 102)(28 47 87 103)(29 48 88 104)(30 49 89 105)(31 50 90 106)(32 51 91 107)
(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)
G:=sub<Sym(128)| (1,52,92,108)(2,53,93,109)(3,54,94,110)(4,55,95,111)(5,56,96,112)(6,57,65,113)(7,58,66,114)(8,59,67,115)(9,60,68,116)(10,61,69,117)(11,62,70,118)(12,63,71,119)(13,64,72,120)(14,33,73,121)(15,34,74,122)(16,35,75,123)(17,36,76,124)(18,37,77,125)(19,38,78,126)(20,39,79,127)(21,40,80,128)(22,41,81,97)(23,42,82,98)(24,43,83,99)(25,44,84,100)(26,45,85,101)(27,46,86,102)(28,47,87,103)(29,48,88,104)(30,49,89,105)(31,50,90,106)(32,51,91,107), (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)>;
G:=Group( (1,52,92,108)(2,53,93,109)(3,54,94,110)(4,55,95,111)(5,56,96,112)(6,57,65,113)(7,58,66,114)(8,59,67,115)(9,60,68,116)(10,61,69,117)(11,62,70,118)(12,63,71,119)(13,64,72,120)(14,33,73,121)(15,34,74,122)(16,35,75,123)(17,36,76,124)(18,37,77,125)(19,38,78,126)(20,39,79,127)(21,40,80,128)(22,41,81,97)(23,42,82,98)(24,43,83,99)(25,44,84,100)(26,45,85,101)(27,46,86,102)(28,47,87,103)(29,48,88,104)(30,49,89,105)(31,50,90,106)(32,51,91,107), (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) );
G=PermutationGroup([[(1,52,92,108),(2,53,93,109),(3,54,94,110),(4,55,95,111),(5,56,96,112),(6,57,65,113),(7,58,66,114),(8,59,67,115),(9,60,68,116),(10,61,69,117),(11,62,70,118),(12,63,71,119),(13,64,72,120),(14,33,73,121),(15,34,74,122),(16,35,75,123),(17,36,76,124),(18,37,77,125),(19,38,78,126),(20,39,79,127),(21,40,80,128),(22,41,81,97),(23,42,82,98),(24,43,83,99),(25,44,84,100),(26,45,85,101),(27,46,86,102),(28,47,87,103),(29,48,88,104),(30,49,89,105),(31,50,90,106),(32,51,91,107)], [(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)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8P | 16A | ··· | 16AF | 32A | ··· | 32BL |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | C16 | C16 | C32 |
| kernel | C4×C32 | C4×C16 | C2×C32 | C32 | C4×C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 4 | 4 | 16 | 16 | 64 |
Matrix representation of C4×C32 ►in GL2(𝔽97) generated by
| 1 | 0 |
| 0 | 22 |
| 46 | 0 |
| 0 | 42 |
G:=sub<GL(2,GF(97))| [1,0,0,22],[46,0,0,42] >;
C4×C32 in GAP, Magma, Sage, TeX
C_4\times C_{32} % in TeX
G:=Group("C4xC32"); // GroupNames label
G:=SmallGroup(128,128);
// by ID
G=gap.SmallGroup(128,128);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,28,64,100,102,124]);
// Polycyclic
G:=Group<a,b|a^4=b^32=1,a*b=b*a>;
// generators/relations
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