direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×S3×Dic3, (C3×S3)⋊C12, C6.25S32, D6.(C3×S3), C3⋊3(S3×C12), C6.1(S3×C6), C33⋊4(C2×C4), (S3×C6).6S3, (S3×C6).3C6, C3⋊Dic3⋊4C6, C3⋊1(C6×Dic3), (C3×C6).38D6, (S3×C32)⋊2C4, C32⋊4(C2×C12), C32⋊12(C4×S3), (C3×Dic3)⋊2C6, C32⋊7(C2×Dic3), (C32×Dic3)⋊3C2, (C32×C6).1C22, C2.1(C3×S32), (S3×C3×C6).1C2, (C3×C6).6(C2×C6), (C3×C3⋊Dic3)⋊1C2, SmallGroup(216,119)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×S3×Dic3 |
Generators and relations for C3×S3×Dic3
G = < a,b,c,d,e | a3=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 228 in 90 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C3×S3, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C32, C32×C6, S3×Dic3, S3×C12, C6×Dic3, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C3×S3×Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S32, S3×C6, S3×Dic3, S3×C12, C6×Dic3, C3×S32, C3×S3×Dic3
►On 24 points - transitive group 24T545
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 8 4 11)(2 7 5 10)(3 12 6 9)(13 20 16 23)(14 19 17 22)(15 24 18 21)
G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,20,16,23)(14,19,17,22)(15,24,18,21)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,20,16,23)(14,19,17,22)(15,24,18,21) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,22),(8,23),(9,24),(10,19),(11,20),(12,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,8,4,11),(2,7,5,10),(3,12,6,9),(13,20,16,23),(14,19,17,22),(15,24,18,21)]])
G:=TransitiveGroup(24,545);
C3×S3×Dic3 is a maximal subgroup of
(S3×C6).D6 D6.S32 D6.4S32 D6.3S32 S32×C12
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | ··· | 6U | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 3 | 3 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | S3 | Dic3 | D6 | C3×S3 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C12 | S32 | S3×Dic3 | C3×S32 | C3×S3×Dic3 |
| kernel | C3×S3×Dic3 | C32×Dic3 | C3×C3⋊Dic3 | S3×C3×C6 | S3×Dic3 | S3×C32 | C3×Dic3 | C3⋊Dic3 | S3×C6 | C3×S3 | C3×Dic3 | S3×C6 | C3×S3 | C3×C6 | Dic3 | D6 | C32 | S3 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×S3×Dic3 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 0 | 3 | 1 |
| 5 | 3 | 6 | 0 |
| 1 | 6 | 4 | 5 |
| 3 | 3 | 2 | 6 |
| 4 | 2 | 0 | 6 |
| 1 | 3 | 3 | 2 |
| 5 | 6 | 0 | 3 |
| 3 | 0 | 6 | 0 |
| 5 | 1 | 2 | 1 |
| 3 | 4 | 6 | 4 |
| 6 | 6 | 5 | 2 |
| 6 | 1 | 4 | 2 |
| 4 | 0 | 5 | 0 |
| 4 | 6 | 4 | 6 |
| 5 | 0 | 3 | 0 |
| 4 | 2 | 0 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,1,5,3,2,3,6,0,0,3,0,6,6,2,3,0],[5,3,6,6,1,4,6,1,2,6,5,4,1,4,2,2],[4,4,5,4,0,6,0,2,5,4,3,0,0,6,0,1] >;
C3×S3×Dic3 in GAP, Magma, Sage, TeX
C_3\times S_3\times {\rm Dic}_3 % in TeX
G:=Group("C3xS3xDic3"); // GroupNames label
G:=SmallGroup(216,119);
// by ID
G=gap.SmallGroup(216,119);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,730,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations