direct product, metacyclic, supersoluble, monomial
Aliases: C3×C4.Dic3, C12.1C12, C12.67D6, C62.5C4, C12.9Dic3, C32⋊7M4(2), C3⋊C8⋊5C6, C4.(C3×Dic3), C4.15(S3×C6), (C2×C6).6C12, C6.6(C2×C12), (C3×C12).5C4, (C6×C12).7C2, (C2×C12).6C6, (C2×C12).18S3, C12.16(C2×C6), C3⋊2(C3×M4(2)), (C2×C6).2Dic3, C2.3(C6×Dic3), C22.(C3×Dic3), C6.18(C2×Dic3), (C3×C12).45C22, (C3×C3⋊C8)⋊12C2, (C2×C4).2(C3×S3), (C3×C6).28(C2×C4), SmallGroup(144,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.Dic3
G = < a,b,c,d | a3=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Permutation representations of C3×C4.Dic3
►On 24 points - transitive group 24T211
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 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 23 10 20 7 17 4 14)(2 16 11 13 8 22 5 19)(3 21 12 18 9 15 6 24)
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,23,10,20,7,17,4,14)(2,16,11,13,8,22,5,19)(3,21,12,18,9,15,6,24)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,23,10,20,7,17,4,14)(2,16,11,13,8,22,5,19)(3,21,12,18,9,15,6,24) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,23,10,20,7,17,4,14),(2,16,11,13,8,22,5,19),(3,21,12,18,9,15,6,24)]])
G:=TransitiveGroup(24,211);
C3×C4.Dic3 is a maximal subgroup of
C12.D12 C12.70D12 C12.14D12 C12.71D12 D12⋊4Dic3 C12.80D12 C12.82D12 C62.5Q8 D12.Dic3 C3⋊C8.22D6 C3⋊C8⋊20D6 D12⋊18D6 D12.28D6 D12.29D6 Dic6.29D6 C3×S3×M4(2) He3⋊7M4(2) C36.C12 He3⋊8M4(2)
C3×C4.Dic3 is a maximal quotient of
He3⋊7M4(2) C36.C12
54 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6M | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | Dic3 | M4(2) | C3×S3 | C3×Dic3 | S3×C6 | C3×Dic3 | C4.Dic3 | C3×M4(2) | C3×C4.Dic3 |
| kernel | C3×C4.Dic3 | C3×C3⋊C8 | C6×C12 | C4.Dic3 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C2×C12 | C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C4.Dic3 ►in GL2(𝔽13) generated by
| 3 | 0 |
| 0 | 3 |
| 8 | 0 |
| 0 | 5 |
| 2 | 0 |
| 0 | 6 |
| 0 | 5 |
| 1 | 0 |
G:=sub<GL(2,GF(13))| [3,0,0,3],[8,0,0,5],[2,0,0,6],[0,1,5,0] >;
C3×C4.Dic3 in GAP, Magma, Sage, TeX
C_3\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C3xC4.Dic3"); // GroupNames label
G:=SmallGroup(144,75);
// by ID
G=gap.SmallGroup(144,75);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,69,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations
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