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G = C3×D6⋊S3order 216 = 23·33

Direct product of C3 and D6⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×D6⋊S3, C334D4, C6.27S32, (S3×C6)⋊1S3, (S3×C6)⋊1C6, D61(C3×S3), C6.3(S3×C6), C3⋊Dic35C6, (C3×C6).40D6, C324(C3×D4), C3211(C3⋊D4), (C32×C6).3C22, (S3×C3×C6)⋊1C2, C2.3(C3×S32), C32(C3×C3⋊D4), (C3×C6).8(C2×C6), (C3×C3⋊Dic3)⋊7C2, SmallGroup(216,121)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×D6⋊S3
C1C3C32C3×C6C32×C6S3×C3×C6 — C3×D6⋊S3
C32C3×C6 — C3×D6⋊S3
C1C6

Generators and relations for C3×D6⋊S3
 G = < a,b,c,d,e | a3=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 276 in 94 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C3 [×4], C4, C22 [×2], S3 [×2], C6, C6 [×2], C6 [×12], D4, C32, C32 [×2], C32 [×4], Dic3 [×3], C12, D6 [×2], C2×C6 [×6], C3×S3 [×8], C3×C6, C3×C6 [×2], C3×C6 [×6], C3⋊D4 [×2], C3×D4, C33, C3×Dic3 [×3], C3⋊Dic3, S3×C6 [×4], S3×C6 [×2], C62 [×2], S3×C32 [×2], C32×C6, D6⋊S3, C3×C3⋊D4 [×2], C3×C3⋊Dic3, S3×C3×C6 [×2], C3×D6⋊S3
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D4, D6 [×2], C2×C6, C3×S3 [×2], C3⋊D4 [×2], C3×D4, S32, S3×C6 [×2], D6⋊S3, C3×C3⋊D4 [×2], C3×S32, C3×D6⋊S3

Permutation representations of C3×D6⋊S3
On 24 points - transitive group 24T546
Generators in S24
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(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)

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

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

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

G:=TransitiveGroup(24,546);

C3×D6⋊S3 is a maximal subgroup of
C33⋊D8  C337SD16  D64S32  (S3×C6)⋊D6  (S3×C6).D6  D6.4S32  D6⋊S3⋊S3  C3×S3×C3⋊D4

45 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K 4 6A6B6C···6H6I6J6K6L···6AA12A12B
order1222333···33334666···66666···61212
size1166112···244418112···24446···61818

45 irreducible representations

dim111111222222224444
type+++++++-
imageC1C2C2C3C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4S32D6⋊S3C3×S32C3×D6⋊S3
kernelC3×D6⋊S3C3×C3⋊Dic3S3×C3×C6D6⋊S3C3⋊Dic3S3×C6S3×C6C33C3×C6D6C32C32C6C3C6C3C2C1
# reps112224212442481122

Matrix representation of C3×D6⋊S3 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
0031
5460
1655
3320
,
3350
6046
6132
0001
,
2656
4313
1125
1635
,
1420
5331
4605
3323
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,5,1,3,0,4,6,3,3,6,5,2,1,0,5,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1],[2,4,1,1,6,3,1,6,5,1,2,3,6,3,5,5],[1,5,4,3,4,3,6,3,2,3,0,2,0,1,5,3] >;

C3×D6⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes S_3
% in TeX

G:=Group("C3xD6:S3");
// GroupNames label

G:=SmallGroup(216,121);
// by ID

G=gap.SmallGroup(216,121);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,730,5189]);
// Polycyclic

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

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