direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C3×D32, C96⋊3C2, C32⋊1C6, D16⋊1C6, C24.64D4, C6.15D16, C12.39D8, C48.19C22, C8.5(C3×D4), C4.1(C3×D8), (C3×D16)⋊5C2, C16.2(C2×C6), C2.3(C3×D16), SmallGroup(192,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D32
G = < a,b,c | a3=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D32
►On 96 points
Generators in S96
(1 58 92)(2 59 93)(3 60 94)(4 61 95)(5 62 96)(6 63 65)(7 64 66)(8 33 67)(9 34 68)(10 35 69)(11 36 70)(12 37 71)(13 38 72)(14 39 73)(15 40 74)(16 41 75)(17 42 76)(18 43 77)(19 44 78)(20 45 79)(21 46 80)(22 47 81)(23 48 82)(24 49 83)(25 50 84)(26 51 85)(27 52 86)(28 53 87)(29 54 88)(30 55 89)(31 56 90)(32 57 91)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(87 96)(88 95)(89 94)(90 93)(91 92)
G:=sub<Sym(96)| (1,58,92)(2,59,93)(3,60,94)(4,61,95)(5,62,96)(6,63,65)(7,64,66)(8,33,67)(9,34,68)(10,35,69)(11,36,70)(12,37,71)(13,38,72)(14,39,73)(15,40,74)(16,41,75)(17,42,76)(18,43,77)(19,44,78)(20,45,79)(21,46,80)(22,47,81)(23,48,82)(24,49,83)(25,50,84)(26,51,85)(27,52,86)(28,53,87)(29,54,88)(30,55,89)(31,56,90)(32,57,91), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(87,96)(88,95)(89,94)(90,93)(91,92)>;
G:=Group( (1,58,92)(2,59,93)(3,60,94)(4,61,95)(5,62,96)(6,63,65)(7,64,66)(8,33,67)(9,34,68)(10,35,69)(11,36,70)(12,37,71)(13,38,72)(14,39,73)(15,40,74)(16,41,75)(17,42,76)(18,43,77)(19,44,78)(20,45,79)(21,46,80)(22,47,81)(23,48,82)(24,49,83)(25,50,84)(26,51,85)(27,52,86)(28,53,87)(29,54,88)(30,55,89)(31,56,90)(32,57,91), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(87,96)(88,95)(89,94)(90,93)(91,92) );
G=PermutationGroup([[(1,58,92),(2,59,93),(3,60,94),(4,61,95),(5,62,96),(6,63,65),(7,64,66),(8,33,67),(9,34,68),(10,35,69),(11,36,70),(12,37,71),(13,38,72),(14,39,73),(15,40,74),(16,41,75),(17,42,76),(18,43,77),(19,44,78),(20,45,79),(21,46,80),(22,47,81),(23,48,82),(24,49,83),(25,50,84),(26,51,85),(27,52,86),(28,53,87),(29,54,88),(30,55,89),(31,56,90),(32,57,91)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(87,96),(88,95),(89,94),(90,93),(91,92)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 32A | ··· | 32H | 48A | ··· | 48H | 96A | ··· | 96P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 16 | 16 | 1 | 1 | 2 | 1 | 1 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | D16 | C3×D8 | D32 | C3×D16 | C3×D32 |
| kernel | C3×D32 | C96 | C3×D16 | D32 | C32 | D16 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×D32 ►in GL2(𝔽31) generated by
| 25 | 0 |
| 0 | 25 |
| 0 | 6 |
| 5 | 10 |
| 1 | 2 |
| 0 | 30 |
G:=sub<GL(2,GF(31))| [25,0,0,25],[0,5,6,10],[1,0,2,30] >;
C3×D32 in GAP, Magma, Sage, TeX
C_3\times D_{32} % in TeX
G:=Group("C3xD32"); // GroupNames label
G:=SmallGroup(192,177);
// by ID
G=gap.SmallGroup(192,177);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,197,1011,514,192,2524,1271,242,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export