direct product, metabelian, supersoluble, monomial
Aliases: C3×D6.4D6, C62.88D6, D6.4(S3×C6), D6⋊S3⋊4C6, (S3×Dic3)⋊2C6, (S3×C6).24D6, C32⋊2Q8⋊5C6, C33⋊22(C4○D4), C62.24(C2×C6), Dic3.4(S3×C6), (C3×Dic3).27D6, (C3×C62).18C22, (C32×C6).31C23, C32⋊25(D4⋊2S3), (C32×Dic3).12C22, C2.13(S32×C6), (C2×C6).46S32, C6.12(S3×C2×C6), C6.115(C2×S32), (C3×C3⋊D4)⋊5S3, C3⋊D4⋊1(C3×S3), (C3×C3⋊D4)⋊2C6, C22.2(C3×S32), (C3×S3×Dic3)⋊7C2, C3⋊4(C3×D4⋊2S3), (S3×C6).4(C2×C6), (C2×C6).15(S3×C6), C32⋊9(C3×C4○D4), (C2×C3⋊Dic3)⋊11C6, (C6×C3⋊Dic3)⋊15C2, (C32×C3⋊D4)⋊2C2, (S3×C3×C6).13C22, (C3×D6⋊S3)⋊11C2, (C3×C32⋊2Q8)⋊11C2, C3⋊Dic3.19(C2×C6), (C3×C6).22(C22×C6), (C3×Dic3).5(C2×C6), (C3×C6).136(C22×S3), (C3×C3⋊Dic3).56C22, SmallGroup(432,653)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6.4D6
G = < a,b,c,d,e | a3=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d5 >
Subgroups: 672 in 218 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C3 [×4], C4 [×4], C22, C22 [×2], S3 [×2], C6, C6 [×2], C6 [×19], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×2], C32 [×4], Dic3 [×2], Dic3 [×6], C12 [×8], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×10], C4○D4, C3×S3 [×6], C3×C6, C3×C6 [×2], C3×C6 [×15], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×5], C3⋊D4 [×2], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×7], C3×Q8, C33, C3×Dic3 [×4], C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×2], C62, C62 [×2], C62 [×6], D4⋊2S3 [×2], C3×C4○D4, S3×C32 [×2], C32×C6, C32×C6, S3×Dic3 [×2], D6⋊S3, C32⋊2Q8, C3×Dic6 [×2], S3×C12 [×2], C6×Dic3 [×5], C3×C3⋊D4 [×4], C3×C3⋊D4 [×4], C2×C3⋊Dic3, D4×C32 [×2], C32×Dic3 [×2], C3×C3⋊Dic3 [×2], S3×C3×C6 [×2], C3×C62, D6.4D6, C3×D4⋊2S3 [×2], C3×S3×Dic3 [×2], C3×D6⋊S3, C3×C32⋊2Q8, C32×C3⋊D4 [×2], C6×C3⋊Dic3, C3×D6.4D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3 [×2], C6 [×7], C23, D6 [×6], C2×C6 [×7], C4○D4, C3×S3 [×2], C22×S3 [×2], C22×C6, S32, S3×C6 [×6], D4⋊2S3 [×2], C3×C4○D4, C2×S32, S3×C2×C6 [×2], C3×S32, D6.4D6, C3×D4⋊2S3 [×2], S32×C6, C3×D6.4D6
►On 24 points - transitive group 24T1283
(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 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 24)(14 15)(16 17)(18 19)(20 21)(22 23)
(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 20 7 14)(2 13 8 19)(3 18 9 24)(4 23 10 17)(5 16 11 22)(6 21 12 15)
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), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,24)(14,15)(16,17)(18,19)(20,21)(22,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,20,7,14)(2,13,8,19)(3,18,9,24)(4,23,10,17)(5,16,11,22)(6,21,12,15)>;
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), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,15,17,19,21,23)(14,24,22,20,18,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,24)(14,15)(16,17)(18,19)(20,21)(22,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,20,7,14)(2,13,8,19)(3,18,9,24)(4,23,10,17)(5,16,11,22)(6,21,12,15) );
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)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,15,17,19,21,23),(14,24,22,20,18,16)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,24),(14,15),(16,17),(18,19),(20,21),(22,23)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,20,7,14),(2,13,8,19),(3,18,9,24),(4,23,10,17),(5,16,11,22),(6,21,12,15)])
G:=TransitiveGroup(24,1283);
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6J | 6K | ··· | 6Y | 6Z | 6AA | 6AB | 6AC | 6AD | ··· | 6AI | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12N | 12O | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 9 | 9 | 18 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | 18 |
72 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | S3×C6 | C3×C4○D4 | S32 | D4⋊2S3 | C2×S32 | C3×S32 | D6.4D6 | C3×D4⋊2S3 | S32×C6 | C3×D6.4D6 |
| kernel | C3×D6.4D6 | C3×S3×Dic3 | C3×D6⋊S3 | C3×C32⋊2Q8 | C32×C3⋊D4 | C6×C3⋊Dic3 | D6.4D6 | S3×Dic3 | D6⋊S3 | C32⋊2Q8 | C3×C3⋊D4 | C2×C3⋊Dic3 | C3×C3⋊D4 | C3×Dic3 | S3×C6 | C62 | C33 | C3⋊D4 | Dic3 | D6 | C2×C6 | C32 | C2×C6 | C32 | C6 | C22 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C3×D6.4D6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 5 | 6 |
| 4 | 4 | 1 | 3 |
| 1 | 1 | 3 | 5 |
| 1 | 6 | 3 | 6 |
| 5 | 1 | 2 | 2 |
| 6 | 0 | 1 | 2 |
| 0 | 0 | 1 | 0 |
| 6 | 1 | 4 | 1 |
| 2 | 3 | 1 | 3 |
| 2 | 5 | 1 | 4 |
| 1 | 4 | 2 | 3 |
| 1 | 1 | 1 | 5 |
| 3 | 0 | 2 | 6 |
| 1 | 3 | 2 | 1 |
| 3 | 1 | 0 | 2 |
| 2 | 2 | 6 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,4,1,1,6,4,1,6,5,1,3,3,6,3,5,6],[5,6,0,6,1,0,0,1,2,1,1,4,2,2,0,1],[2,2,1,1,3,5,4,1,1,1,2,1,3,4,3,5],[3,1,3,2,0,3,1,2,2,2,0,6,6,1,2,1] >;
C3×D6.4D6 in GAP, Magma, Sage, TeX
C_3\times D_6._4D_6
% in TeX
G:=Group("C3xD6.4D6"); // GroupNames label
G:=SmallGroup(432,653);
// by ID
G=gap.SmallGroup(432,653);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^2=1,d^6=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^4*c,e*d*e^-1=d^5>;
// generators/relations