direct product, metabelian, supersoluble, monomial
Aliases: C3×D12⋊6C22, C62.122D4, D4⋊S3⋊5C6, (C6×D4)⋊9S3, (C6×D4)⋊2C6, C4○D12⋊3C6, D12⋊6(C2×C6), D4.S3⋊5C6, D4.6(S3×C6), C6.45(C6×D4), Dic6⋊5(C2×C6), (C3×D4).46D6, C12.15(C3×D4), (C3×C12).85D4, C4.Dic3⋊6C6, (C2×C12).240D6, (C3×D12)⋊24C22, C32⋊22(C8⋊C22), (C3×C12).83C23, C12.12(C22×C6), C12.103(C3⋊D4), (C6×C12).118C22, C12.163(C22×S3), (C3×Dic6)⋊22C22, (D4×C32).22C22, C3⋊C8⋊3(C2×C6), (D4×C3×C6)⋊2C2, C4.12(S3×C2×C6), (C2×D4)⋊2(C3×S3), C3⋊4(C3×C8⋊C22), C2.9(C6×C3⋊D4), (C3×D4⋊S3)⋊13C2, (C3×C4○D12)⋊7C2, (C3×C3⋊C8)⋊20C22, (C2×C4).15(S3×C6), (C3×D4).6(C2×C6), (C2×C6).48(C3×D4), C4.16(C3×C3⋊D4), (C2×C12).29(C2×C6), (C3×D4.S3)⋊11C2, (C3×C6).255(C2×D4), C6.146(C2×C3⋊D4), (C3×C4.Dic3)⋊5C2, (C2×C6).63(C3⋊D4), C22.10(C3×C3⋊D4), SmallGroup(288,703)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D12⋊6C22
G = < a,b,c,d,e | a3=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >
Subgroups: 394 in 163 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C8⋊C22, C3×Dic3, C3×C12, S3×C6, C62, C62, C4.Dic3, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C3×SD16, C4○D12, C6×D4, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D4×C32, D4×C32, C2×C62, D12⋊6C22, C3×C8⋊C22, C3×C4.Dic3, C3×D4⋊S3, C3×D4.S3, C3×C4○D12, D4×C3×C6, C3×D12⋊6C22
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C8⋊C22, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, D12⋊6C22, C3×C8⋊C22, C6×C3⋊D4, C3×D12⋊6C22
►On 24 points - transitive group 24T628
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 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)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)
(2 8)(4 10)(6 12)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)
G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (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)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (2,8)(4,10)(6,12)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (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)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (2,8)(4,10)(6,12)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(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),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)], [(2,8),(4,10),(6,12),(13,16),(14,23),(15,18),(17,20),(19,22),(21,24)]])
G:=TransitiveGroup(24,628);
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6M | 6N | ··· | 6AC | 6AD | 6AE | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C3×C3⋊D4 | C3×C3⋊D4 | C8⋊C22 | D12⋊6C22 | C3×C8⋊C22 | C3×D12⋊6C22 |
| kernel | C3×D12⋊6C22 | C3×C4.Dic3 | C3×D4⋊S3 | C3×D4.S3 | C3×C4○D12 | D4×C3×C6 | D12⋊6C22 | C4.Dic3 | D4⋊S3 | D4.S3 | C4○D12 | C6×D4 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C2×D4 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | D4 | C4 | C22 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×D12⋊6C22 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 6 | 2 |
| 2 | 1 | 0 | 1 |
| 5 | 5 | 5 | 4 |
| 4 | 3 | 5 | 0 |
| 6 | 2 | 5 | 1 |
| 1 | 1 | 2 | 1 |
| 4 | 3 | 5 | 5 |
| 6 | 6 | 4 | 2 |
| 0 | 1 | 4 | 5 |
| 1 | 0 | 3 | 5 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,2,5,4,0,1,5,3,6,0,5,5,2,1,4,0],[6,1,4,6,2,1,3,6,5,2,5,4,1,1,5,2],[0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[0,1,0,0,1,0,0,0,1,1,6,0,0,0,0,1] >;
C3×D12⋊6C22 in GAP, Magma, Sage, TeX
C_3\times D_{12}\rtimes_6C_2^2 % in TeX
G:=Group("C3xD12:6C2^2"); // GroupNames label
G:=SmallGroup(288,703);
// by ID
G=gap.SmallGroup(288,703);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,2524,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^2=e^2=1,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,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations