metabelian, soluble, monomial, A-group
Aliases: C32:2C32, (C3xC12).4C8, (C3xC24).1C4, (C3xC6).2C16, C8.4(C32:C4), C2.(C32:2C16), C24.S3.3C2, C4.2(C32:2C8), SmallGroup(288,188)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32:2C32 |
Generators and relations for C32:2C32
G = < a,b,c | a3=b3=c32=1, cbc-1=ab=ba, cac-1=a-1b >
Quotients: C1, C2, C4, C8, C16, C32, C32:C4, C32:2C8, C32:2C16, C32:2C32
Smallest permutation representation of C32:2C32
►On 96 points
Generators in S96
(2 40 70)(4 72 42)(6 44 74)(8 76 46)(10 48 78)(12 80 50)(14 52 82)(16 84 54)(18 56 86)(20 88 58)(22 60 90)(24 92 62)(26 64 94)(28 96 34)(30 36 66)(32 68 38)
(1 39 69)(2 40 70)(3 71 41)(4 72 42)(5 43 73)(6 44 74)(7 75 45)(8 76 46)(9 47 77)(10 48 78)(11 79 49)(12 80 50)(13 51 81)(14 52 82)(15 83 53)(16 84 54)(17 55 85)(18 56 86)(19 87 57)(20 88 58)(21 59 89)(22 60 90)(23 91 61)(24 92 62)(25 63 93)(26 64 94)(27 95 33)(28 96 34)(29 35 65)(30 36 66)(31 67 37)(32 68 38)
(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)| (2,40,70)(4,72,42)(6,44,74)(8,76,46)(10,48,78)(12,80,50)(14,52,82)(16,84,54)(18,56,86)(20,88,58)(22,60,90)(24,92,62)(26,64,94)(28,96,34)(30,36,66)(32,68,38), (1,39,69)(2,40,70)(3,71,41)(4,72,42)(5,43,73)(6,44,74)(7,75,45)(8,76,46)(9,47,77)(10,48,78)(11,79,49)(12,80,50)(13,51,81)(14,52,82)(15,83,53)(16,84,54)(17,55,85)(18,56,86)(19,87,57)(20,88,58)(21,59,89)(22,60,90)(23,91,61)(24,92,62)(25,63,93)(26,64,94)(27,95,33)(28,96,34)(29,35,65)(30,36,66)(31,67,37)(32,68,38), (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( (2,40,70)(4,72,42)(6,44,74)(8,76,46)(10,48,78)(12,80,50)(14,52,82)(16,84,54)(18,56,86)(20,88,58)(22,60,90)(24,92,62)(26,64,94)(28,96,34)(30,36,66)(32,68,38), (1,39,69)(2,40,70)(3,71,41)(4,72,42)(5,43,73)(6,44,74)(7,75,45)(8,76,46)(9,47,77)(10,48,78)(11,79,49)(12,80,50)(13,51,81)(14,52,82)(15,83,53)(16,84,54)(17,55,85)(18,56,86)(19,87,57)(20,88,58)(21,59,89)(22,60,90)(23,91,61)(24,92,62)(25,63,93)(26,64,94)(27,95,33)(28,96,34)(29,35,65)(30,36,66)(31,67,37)(32,68,38), (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([[(2,40,70),(4,72,42),(6,44,74),(8,76,46),(10,48,78),(12,80,50),(14,52,82),(16,84,54),(18,56,86),(20,88,58),(22,60,90),(24,92,62),(26,64,94),(28,96,34),(30,36,66),(32,68,38)], [(1,39,69),(2,40,70),(3,71,41),(4,72,42),(5,43,73),(6,44,74),(7,75,45),(8,76,46),(9,47,77),(10,48,78),(11,79,49),(12,80,50),(13,51,81),(14,52,82),(15,83,53),(16,84,54),(17,55,85),(18,56,86),(19,87,57),(20,88,58),(21,59,89),(22,60,90),(23,91,61),(24,92,62),(25,63,93),(26,64,94),(27,95,33),(28,96,34),(29,35,65),(30,36,66),(31,67,37),(32,68,38)], [(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)]])
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 32A | ··· | 32P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 32 | ··· | 32 |
| size | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C32:C4 | C32:2C8 | C32:2C16 | C32:2C32 |
| kernel | C32:2C32 | C24.S3 | C3xC24 | C3xC12 | C3xC6 | C32 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of C32:2C32 ►in GL5(F97)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 96 | 96 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 96 | 96 |
| 0 | 0 | 0 | 1 | 0 |
| 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 60 | 6 | 0 | 0 |
| 0 | 43 | 37 | 0 | 0 |
G:=sub<GL(5,GF(97))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,96,1,0,0,0,96,0],[1,0,0,0,0,0,0,96,0,0,0,1,96,0,0,0,0,0,96,1,0,0,0,96,0],[63,0,0,0,0,0,0,0,60,43,0,0,0,6,37,0,1,0,0,0,0,0,1,0,0] >;
C32:2C32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2C_{32} % in TeX
G:=Group("C3^2:2C32"); // GroupNames label
G:=SmallGroup(288,188);
// by ID
G=gap.SmallGroup(288,188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,14,36,58,80,9413,1356,12550,4717]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^32=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b>;
// generators/relations
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