metabelian, soluble, monomial, A-group
Aliases: C32⋊C32, C4.2F9, (C3×C6).C16, C2.(C2.F9), (C3×C12).1C8, C32⋊4C8.C4, C32⋊2C16.1C2, SmallGroup(288,373)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊C32 |
Generators and relations for C32⋊C32
G = < a,b,c | a3=b3=c32=1, cac-1=ab=ba, cbc-1=a >
Smallest permutation representation of C32⋊C32
►On 96 points
Generators in S96
(2 79 55)(3 80 56)(4 57 81)(6 59 83)(7 60 84)(8 85 61)(10 87 63)(11 88 64)(12 33 89)(14 35 91)(15 36 92)(16 93 37)(18 95 39)(19 96 40)(20 41 65)(22 43 67)(23 44 68)(24 69 45)(26 71 47)(27 72 48)(28 49 73)(30 51 75)(31 52 76)(32 77 53)
(1 78 54)(3 80 56)(4 81 57)(5 58 82)(7 60 84)(8 61 85)(9 86 62)(11 88 64)(12 89 33)(13 34 90)(15 36 92)(16 37 93)(17 94 38)(19 96 40)(20 65 41)(21 42 66)(23 44 68)(24 45 69)(25 70 46)(27 72 48)(28 73 49)(29 50 74)(31 52 76)(32 53 77)
(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)
G:=sub<Sym(96)| (2,79,55)(3,80,56)(4,57,81)(6,59,83)(7,60,84)(8,85,61)(10,87,63)(11,88,64)(12,33,89)(14,35,91)(15,36,92)(16,93,37)(18,95,39)(19,96,40)(20,41,65)(22,43,67)(23,44,68)(24,69,45)(26,71,47)(27,72,48)(28,49,73)(30,51,75)(31,52,76)(32,77,53), (1,78,54)(3,80,56)(4,81,57)(5,58,82)(7,60,84)(8,61,85)(9,86,62)(11,88,64)(12,89,33)(13,34,90)(15,36,92)(16,37,93)(17,94,38)(19,96,40)(20,65,41)(21,42,66)(23,44,68)(24,45,69)(25,70,46)(27,72,48)(28,73,49)(29,50,74)(31,52,76)(32,53,77), (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)>;
G:=Group( (2,79,55)(3,80,56)(4,57,81)(6,59,83)(7,60,84)(8,85,61)(10,87,63)(11,88,64)(12,33,89)(14,35,91)(15,36,92)(16,93,37)(18,95,39)(19,96,40)(20,41,65)(22,43,67)(23,44,68)(24,69,45)(26,71,47)(27,72,48)(28,49,73)(30,51,75)(31,52,76)(32,77,53), (1,78,54)(3,80,56)(4,81,57)(5,58,82)(7,60,84)(8,61,85)(9,86,62)(11,88,64)(12,89,33)(13,34,90)(15,36,92)(16,37,93)(17,94,38)(19,96,40)(20,65,41)(21,42,66)(23,44,68)(24,45,69)(25,70,46)(27,72,48)(28,73,49)(29,50,74)(31,52,76)(32,53,77), (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) );
G=PermutationGroup([[(2,79,55),(3,80,56),(4,57,81),(6,59,83),(7,60,84),(8,85,61),(10,87,63),(11,88,64),(12,33,89),(14,35,91),(15,36,92),(16,93,37),(18,95,39),(19,96,40),(20,41,65),(22,43,67),(23,44,68),(24,69,45),(26,71,47),(27,72,48),(28,49,73),(30,51,75),(31,52,76),(32,77,53)], [(1,78,54),(3,80,56),(4,81,57),(5,58,82),(7,60,84),(8,61,85),(9,86,62),(11,88,64),(12,89,33),(13,34,90),(15,36,92),(16,37,93),(17,94,38),(19,96,40),(20,65,41),(21,42,66),(23,44,68),(24,45,69),(25,70,46),(27,72,48),(28,73,49),(29,50,74),(31,52,76),(32,53,77)], [(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)]])
36 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 8 | 1 | 1 | 8 | 9 | 9 | 9 | 9 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | F9 | C2.F9 | C32⋊C32 |
| kernel | C32⋊C32 | C32⋊2C16 | C32⋊4C8 | C3×C12 | C3×C6 | C32 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 1 | 2 |
Matrix representation of C32⋊C32 ►in GL9(𝔽97)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 0 | 1 | 0 | 0 |
| 0 | 64 | 42 | 73 | 0 | 96 | 96 | 0 | 0 |
| 0 | 62 | 19 | 2 | 0 | 0 | 0 | 96 | 96 |
| 0 | 0 | 0 | 0 | 95 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 96 | 0 | 0 | 0 | 0 |
| 0 | 72 | 17 | 73 | 0 | 96 | 96 | 0 | 0 |
| 0 | 8 | 72 | 0 | 24 | 1 | 0 | 0 | 0 |
| 0 | 17 | 79 | 0 | 95 | 0 | 0 | 1 | 0 |
| 0 | 17 | 79 | 0 | 95 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 | 1 | 0 | 0 |
| 0 | 64 | 42 | 73 | 73 | 95 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 | 1 |
| 0 | 62 | 19 | 2 | 2 | 0 | 0 | 95 | 96 |
| 0 | 12 | 48 | 68 | 68 | 80 | 72 | 24 | 0 |
| 0 | 12 | 48 | 69 | 68 | 80 | 72 | 24 | 0 |
| 0 | 34 | 53 | 11 | 11 | 96 | 79 | 95 | 0 |
| 0 | 27 | 44 | 11 | 11 | 96 | 79 | 95 | 0 |
G:=sub<GL(9,GF(97))| [1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,64,62,0,0,0,1,0,0,0,42,19,0,0,0,0,0,1,0,73,2,0,0,0,0,96,96,24,0,0,95,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0],[1,0,0,0,0,0,0,0,0,0,96,96,0,0,72,8,17,17,0,1,0,0,0,17,72,79,79,0,0,0,0,1,73,0,0,0,0,0,0,96,96,0,24,95,95,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,0,0,0,0,64,0,62,12,12,34,27,0,0,42,0,19,48,48,53,44,0,0,73,0,2,68,69,11,11,0,0,73,0,2,68,68,11,11,0,96,95,0,0,80,80,96,96,0,1,96,0,0,72,72,79,79,0,0,0,96,95,24,24,95,95,0,0,0,1,96,0,0,0,0] >;
C32⋊C32 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_{32} % in TeX
G:=Group("C3^2:C32"); // GroupNames label
G:=SmallGroup(288,373);
// by ID
G=gap.SmallGroup(288,373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,14,36,58,80,4037,4716,691,10982,6285,2372]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^32=1,c*a*c^-1=a*b=b*a,c*b*c^-1=a>;
// generators/relations
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