direct product, metabelian, supersoluble, monomial
Aliases: C3×C32⋊2Q8, C33⋊3Q8, C32⋊7Dic6, C6.29S32, C6.5(S3×C6), C3⋊1(C3×Dic6), (C3×C6).42D6, Dic3.(C3×S3), C32⋊3(C3×Q8), C3⋊Dic3.3C6, (C3×Dic3).1C6, (C3×Dic3).6S3, (C32×C6).5C22, (C32×Dic3).2C2, C2.5(C3×S32), (C3×C6).10(C2×C6), (C3×C3⋊Dic3).5C2, SmallGroup(216,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊2Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 180 in 70 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, Q8, C32, C32, C32, Dic3, Dic3, C12, C3×C6, C3×C6, C3×C6, Dic6, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C32×C6, C32⋊2Q8, C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C3×C32⋊2Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, D6, C2×C6, C3×S3, Dic6, C3×Q8, S32, S3×C6, C32⋊2Q8, C3×Dic6, C3×S32, C3×C32⋊2Q8
►On 24 points - transitive group 24T542
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 14 19)(2 20 15)(3 16 17)(4 18 13)(5 10 21)(6 22 11)(7 12 23)(8 24 9)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,14,19)(2,20,15)(3,16,17)(4,18,13)(5,10,21)(6,22,11)(7,12,23)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,14,19)(2,20,15)(3,16,17)(4,18,13)(5,10,21)(6,22,11)(7,12,23)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,14,19),(2,20,15),(3,16,17),(4,18,13),(5,10,21),(6,22,11),(7,12,23),(8,24,9)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
G:=TransitiveGroup(24,542);
C3×C32⋊2Q8 is a maximal subgroup of
C33⋊6SD16 C33⋊Q16 C33⋊5(C2×Q8) C33⋊6(C2×Q8) D6.3S32 D6.6S32 Dic3.S32 C3×S3×Dic6
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 12A | ··· | 12P | 12Q | 12R |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 18 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 18 | 18 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | Q8 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | C3×Dic6 | S32 | C32⋊2Q8 | C3×S32 | C3×C32⋊2Q8 |
| kernel | C3×C32⋊2Q8 | C32×Dic3 | C3×C3⋊Dic3 | C32⋊2Q8 | C3×Dic3 | C3⋊Dic3 | C3×Dic3 | C33 | C3×C6 | Dic3 | C32 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C32⋊2Q8 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 1 | 2 | 1 |
| 3 | 3 | 6 | 4 |
| 6 | 6 | 4 | 2 |
| 6 | 1 | 4 | 1 |
| 0 | 0 | 4 | 6 |
| 2 | 3 | 1 | 0 |
| 6 | 1 | 2 | 2 |
| 4 | 4 | 5 | 0 |
| 4 | 5 | 4 | 6 |
| 6 | 0 | 6 | 0 |
| 3 | 3 | 3 | 1 |
| 3 | 4 | 2 | 0 |
| 2 | 3 | 6 | 6 |
| 1 | 4 | 0 | 6 |
| 2 | 5 | 6 | 0 |
| 6 | 6 | 6 | 2 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,3,6,6,1,3,6,1,2,6,4,4,1,4,2,1],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[4,6,3,3,5,0,3,4,4,6,3,2,6,0,1,0],[2,1,2,6,3,4,5,6,6,0,6,6,6,6,0,2] >;
C3×C32⋊2Q8 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_2Q_8
% in TeX
G:=Group("C3xC3^2:2Q8"); // GroupNames label
G:=SmallGroup(216,123);
// by ID
G=gap.SmallGroup(216,123);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,730,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations