direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.6D6, C62.33D4, (C22×S3)⋊C12, (C2×Dic3)⋊C12, (C6×Dic3)⋊2C4, (C2×C6).44D12, C23.6(S3×C6), C6.43(D6⋊C4), C6.D4⋊1C6, C32⋊6(C23⋊C4), C22.3(S3×C12), C62.36(C2×C4), C22.2(C3×D12), (C22×C6).24D6, (C2×C62).9C22, (S3×C2×C6)⋊2C4, C3⋊1(C3×C23⋊C4), C2.4(C3×D6⋊C4), (C2×C6).1(C3×D4), (C3×C22⋊C4)⋊1S3, C22⋊C4⋊1(C3×S3), (C3×C22⋊C4)⋊1C6, (C2×C6).1(C2×C12), (C2×C6).58(C4×S3), (C6×C3⋊D4).5C2, (C2×C3⋊D4).1C6, C6.2(C3×C22⋊C4), C22.8(C3×C3⋊D4), (C3×C6.D4)⋊3C2, (C2×C6).61(C3⋊D4), (C32×C22⋊C4)⋊1C2, (C22×C6).16(C2×C6), (C3×C6).42(C22⋊C4), SmallGroup(288,240)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.6D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde5 >
Subgroups: 354 in 121 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, C3×S3, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C23⋊C4, C3×Dic3, C3×C12, S3×C6, C62, C62, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C2×C3⋊D4, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, C2×C62, C23.6D6, C3×C23⋊C4, C3×C6.D4, C32×C22⋊C4, C6×C3⋊D4, C3×C23.6D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C23⋊C4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C23.6D6, C3×C23⋊C4, C3×D6⋊C4, C3×C23.6D6
►On 24 points - transitive group 24T587
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 12)(4 8)(6 10)(13 19)(15 21)(17 23)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 19 11 13)(2 18)(3 23 7 17)(4 22)(5 15 9 21)(6 14)(8 16)(10 20)(12 24)
G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,12)(4,8)(6,10)(13,19)(15,21)(17,23), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,19,11,13)(2,18)(3,23,7,17)(4,22)(5,15,9,21)(6,14)(8,16)(10,20)(12,24)>;
G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,12)(4,8)(6,10)(13,19)(15,21)(17,23), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,19,11,13)(2,18)(3,23,7,17)(4,22)(5,15,9,21)(6,14)(8,16)(10,20)(12,24) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(2,12),(4,8),(6,10),(13,19),(15,21),(17,23)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,19,11,13),(2,18),(3,23,7,17),(4,22),(5,15,9,21),(6,14),(8,16),(10,20),(12,24)]])
G:=TransitiveGroup(24,587);
►On 24 points - transitive group 24T629
(1 3 2)(4 6 5)(7 9 11)(8 10 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 11)(2 9)(3 7)(4 12)(5 10)(6 8)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)
(1 6)(2 4)(3 5)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 8 15)(2 20 12 23)(3 16 10 19)(4 14 9 17)(5 22 7 13)(6 18 11 21)
G:=sub<Sym(24)| (1,3,2)(4,6,5)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,11)(2,9)(3,7)(4,12)(5,10)(6,8)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,6)(2,4)(3,5)(7,10)(8,11)(9,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24,8,15)(2,20,12,23)(3,16,10,19)(4,14,9,17)(5,22,7,13)(6,18,11,21)>;
G:=Group( (1,3,2)(4,6,5)(7,9,11)(8,10,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,11)(2,9)(3,7)(4,12)(5,10)(6,8)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23), (1,6)(2,4)(3,5)(7,10)(8,11)(9,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24,8,15)(2,20,12,23)(3,16,10,19)(4,14,9,17)(5,22,7,13)(6,18,11,21) );
G=PermutationGroup([[(1,3,2),(4,6,5),(7,9,11),(8,10,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,11),(2,9),(3,7),(4,12),(5,10),(6,8),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23)], [(1,6),(2,4),(3,5),(7,10),(8,11),(9,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,24,8,15),(2,20,12,23),(3,16,10,19),(4,14,9,17),(5,22,7,13),(6,18,11,21)]])
G:=TransitiveGroup(24,629);
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6Q | 6R | ··· | 6W | 6X | 6Y | 12A | ··· | 12P | 12Q | ··· | 12V |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 4 | ··· | 4 | 12 | ··· | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D4 | D6 | C3×S3 | C4×S3 | D12 | C3⋊D4 | C3×D4 | S3×C6 | S3×C12 | C3×D12 | C3×C3⋊D4 | C23⋊C4 | C23.6D6 | C3×C23⋊C4 | C3×C23.6D6 |
| kernel | C3×C23.6D6 | C3×C6.D4 | C32×C22⋊C4 | C6×C3⋊D4 | C23.6D6 | C6×Dic3 | S3×C2×C6 | C6.D4 | C3×C22⋊C4 | C2×C3⋊D4 | C2×Dic3 | C22×S3 | C3×C22⋊C4 | C62 | C22×C6 | C22⋊C4 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C23.6D6 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 6 | 3 | 2 |
| 6 | 0 | 4 | 2 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 5 | 2 | 6 |
| 1 | 5 | 5 | 3 |
| 0 | 0 | 1 | 0 |
| 5 | 2 | 1 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 3 | 0 | 4 | 1 |
| 3 | 6 | 5 | 2 |
| 1 | 1 | 1 | 5 |
| 0 | 0 | 0 | 4 |
| 2 | 4 | 3 | 3 |
| 0 | 3 | 3 | 4 |
| 1 | 6 | 3 | 4 |
| 4 | 4 | 5 | 6 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[1,1,0,5,5,5,0,2,2,5,1,1,6,3,0,0],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,3,1,0,0,6,1,0,4,5,1,0,1,2,5,4],[2,0,1,4,4,3,6,4,3,3,3,5,3,4,4,6] >;
C3×C23.6D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._6D_6
% in TeX
G:=Group("C3xC2^3.6D6"); // GroupNames label
G:=SmallGroup(288,240);
// by ID
G=gap.SmallGroup(288,240);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,1271,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;
// generators/relations