Copied to
clipboard

## G = (S3×C6)⋊D6order 432 = 24·33

### 1st semidirect product of S3×C6 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — (S3×C6)⋊D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×D6⋊S3 — (S3×C6)⋊D6
 Lower central C33 — C32×C6 — (S3×C6)⋊D6
 Upper central C1 — C2

Generators and relations for (S3×C6)⋊D6
G = < a,b,c,d,e | a6=b3=c2=d6=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=ebe=b-1, bd=db, dcd-1=a3c, ce=ec, ede=d-1 >

Subgroups: 2276 in 306 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22 [×9], S3 [×24], C6, C6 [×2], C6 [×14], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3, Dic3 [×3], C12 [×4], D6 [×2], D6 [×33], C2×C6 [×8], C2×D4, C3×S3 [×16], C3⋊S3 [×16], C3×C6, C3×C6 [×2], C3×C6 [×6], C4×S3 [×4], D12 [×3], C3⋊D4 [×6], C3×D4 [×3], C22×S3 [×8], C33, C3×Dic3 [×2], C3×Dic3 [×4], C3⋊Dic3, C3×C12, S32 [×16], S3×C6 [×4], S3×C6 [×8], C2×C3⋊S3 [×2], C2×C3⋊S3 [×11], C62 [×2], S3×D4 [×3], S3×C32 [×2], C3×C3⋊S3 [×2], C33⋊C2 [×2], C32×C6, C6.D6 [×3], D6⋊S3, D6⋊S3, C3⋊D12 [×2], C3⋊D12 [×2], C3×D12 [×2], C3×C3⋊D4 [×4], C4×C3⋊S3, C2×S32 [×6], C22×C3⋊S3 [×2], C32×Dic3, C3×C3⋊Dic3, S3×C3⋊S3 [×4], S3×C3×C6 [×2], C6×C3⋊S3 [×2], C2×C33⋊C2, D6⋊D6, Dic3⋊D6 [×2], C3×D6⋊S3, C3×C3⋊D12 [×2], C338(C2×C4), C339D4, C2×S3×C3⋊S3 [×2], (S3×C6)⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, C22×S3 [×3], S32 [×3], S3×D4 [×3], C2×S32 [×3], D6⋊D6, Dic3⋊D6 [×2], S33, (S3×C6)⋊D6

Permutation representations of (S3×C6)⋊D6
On 24 points - transitive group 24T1294
Generators in S24
(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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 19 3 23 5 21)(2 24 4 22 6 20)(7 16 11 18 9 14)(8 15 12 17 10 13)
(1 3)(4 6)(8 12)(9 11)(13 17)(14 16)(20 24)(21 23)

G:=sub<Sym(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), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,19,3,23,5,21)(2,24,4,22,6,20)(7,16,11,18,9,14)(8,15,12,17,10,13), (1,3)(4,6)(8,12)(9,11)(13,17)(14,16)(20,24)(21,23)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,19,3,23,5,21)(2,24,4,22,6,20)(7,16,11,18,9,14)(8,15,12,17,10,13), (1,3)(4,6)(8,12)(9,11)(13,17)(14,16)(20,24)(21,23) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,19,3,23,5,21),(2,24,4,22,6,20),(7,16,11,18,9,14),(8,15,12,17,10,13)], [(1,3),(4,6),(8,12),(9,11),(13,17),(14,16),(20,24),(21,23)])

G:=TransitiveGroup(24,1294);

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H ··· 6O 6P 6Q 12A 12B 12C 12D 12E order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 ··· 6 6 6 12 12 12 12 12 size 1 1 6 6 18 18 27 27 2 2 2 4 4 4 8 6 18 2 2 2 4 4 4 8 12 ··· 12 36 36 12 12 12 12 36

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 S32 S32 S3×D4 C2×S32 D6⋊D6 Dic3⋊D6 S33 (S3×C6)⋊D6 kernel (S3×C6)⋊D6 C3×D6⋊S3 C3×C3⋊D12 C33⋊8(C2×C4) C33⋊9D4 C2×S3×C3⋊S3 D6⋊S3 C3⋊D12 C33⋊C2 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 Dic3 D6 C32 C6 C3 C3 C2 C1 # reps 1 1 2 1 1 2 1 2 2 2 1 4 2 1 2 3 3 2 4 1 1

Matrix representation of (S3×C6)⋊D6 in GL8(ℤ)

 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 1
,
 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0
,
 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1

G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1] >;

(S3×C6)⋊D6 in GAP, Magma, Sage, TeX

(S_3\times C_6)\rtimes D_6
% in TeX

G:=Group("(S3xC6):D6");
// GroupNames label

G:=SmallGroup(432,601);
// by ID

G=gap.SmallGroup(432,601);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,254,135,58,298,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽