metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C96⋊2C2, C32⋊2S3, C3⋊1SD64, C8.6D12, C6.2D16, C4.2D24, C2.4D48, D48.1C2, C12.27D8, C24.56D4, C16.14D6, Dic24⋊1C2, C48.15C22, SmallGroup(192,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊S3
G = < a,b,c | a32=b3=c2=1, ab=ba, cac=a15, cbc=b-1 >
Smallest permutation representation of C32⋊S3
►On 96 points
Generators in S96
(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)
(1 56 85)(2 57 86)(3 58 87)(4 59 88)(5 60 89)(6 61 90)(7 62 91)(8 63 92)(9 64 93)(10 33 94)(11 34 95)(12 35 96)(13 36 65)(14 37 66)(15 38 67)(16 39 68)(17 40 69)(18 41 70)(19 42 71)(20 43 72)(21 44 73)(22 45 74)(23 46 75)(24 47 76)(25 48 77)(26 49 78)(27 50 79)(28 51 80)(29 52 81)(30 53 82)(31 54 83)(32 55 84)
(2 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 92)(34 75)(35 90)(36 73)(37 88)(38 71)(39 86)(40 69)(41 84)(42 67)(43 82)(44 65)(45 80)(46 95)(47 78)(48 93)(49 76)(50 91)(51 74)(52 89)(53 72)(54 87)(55 70)(56 85)(57 68)(58 83)(59 66)(60 81)(61 96)(62 79)(63 94)(64 77)
G:=sub<Sym(96)| (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), (1,56,85)(2,57,86)(3,58,87)(4,59,88)(5,60,89)(6,61,90)(7,62,91)(8,63,92)(9,64,93)(10,33,94)(11,34,95)(12,35,96)(13,36,65)(14,37,66)(15,38,67)(16,39,68)(17,40,69)(18,41,70)(19,42,71)(20,43,72)(21,44,73)(22,45,74)(23,46,75)(24,47,76)(25,48,77)(26,49,78)(27,50,79)(28,51,80)(29,52,81)(30,53,82)(31,54,83)(32,55,84), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,92)(34,75)(35,90)(36,73)(37,88)(38,71)(39,86)(40,69)(41,84)(42,67)(43,82)(44,65)(45,80)(46,95)(47,78)(48,93)(49,76)(50,91)(51,74)(52,89)(53,72)(54,87)(55,70)(56,85)(57,68)(58,83)(59,66)(60,81)(61,96)(62,79)(63,94)(64,77)>;
G:=Group( (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), (1,56,85)(2,57,86)(3,58,87)(4,59,88)(5,60,89)(6,61,90)(7,62,91)(8,63,92)(9,64,93)(10,33,94)(11,34,95)(12,35,96)(13,36,65)(14,37,66)(15,38,67)(16,39,68)(17,40,69)(18,41,70)(19,42,71)(20,43,72)(21,44,73)(22,45,74)(23,46,75)(24,47,76)(25,48,77)(26,49,78)(27,50,79)(28,51,80)(29,52,81)(30,53,82)(31,54,83)(32,55,84), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,92)(34,75)(35,90)(36,73)(37,88)(38,71)(39,86)(40,69)(41,84)(42,67)(43,82)(44,65)(45,80)(46,95)(47,78)(48,93)(49,76)(50,91)(51,74)(52,89)(53,72)(54,87)(55,70)(56,85)(57,68)(58,83)(59,66)(60,81)(61,96)(62,79)(63,94)(64,77) );
G=PermutationGroup([[(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)], [(1,56,85),(2,57,86),(3,58,87),(4,59,88),(5,60,89),(6,61,90),(7,62,91),(8,63,92),(9,64,93),(10,33,94),(11,34,95),(12,35,96),(13,36,65),(14,37,66),(15,38,67),(16,39,68),(17,40,69),(18,41,70),(19,42,71),(20,43,72),(21,44,73),(22,45,74),(23,46,75),(24,47,76),(25,48,77),(26,49,78),(27,50,79),(28,51,80),(29,52,81),(30,53,82),(31,54,83),(32,55,84)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26),(33,92),(34,75),(35,90),(36,73),(37,88),(38,71),(39,86),(40,69),(41,84),(42,67),(43,82),(44,65),(45,80),(46,95),(47,78),(48,93),(49,76),(50,91),(51,74),(52,89),(53,72),(54,87),(55,70),(56,85),(57,68),(58,83),(59,66),(60,81),(61,96),(62,79),(63,94),(64,77)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 32A | ··· | 32H | 48A | ··· | 48H | 96A | ··· | 96P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 48 | 2 | 2 | 48 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | D12 | D16 | D24 | SD64 | D48 | C32⋊S3 |
| kernel | C32⋊S3 | C96 | D48 | Dic24 | C32 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C32⋊S3 ►in GL2(𝔽47) generated by
| 40 | 26 |
| 26 | 11 |
| 26 | 7 |
| 7 | 20 |
| 1 | 26 |
| 0 | 46 |
G:=sub<GL(2,GF(47))| [40,26,26,11],[26,7,7,20],[1,0,26,46] >;
C32⋊S3 in GAP, Magma, Sage, TeX
C_{32}\rtimes S_3 % in TeX
G:=Group("C32:S3"); // GroupNames label
G:=SmallGroup(192,8);
// by ID
G=gap.SmallGroup(192,8);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,926,142,1571,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^32=b^3=c^2=1,a*b=b*a,c*a*c=a^15,c*b*c=b^-1>;
// generators/relations
Export