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G = C3xD6:S3order 216 = 23·33

Direct product of C3 and D6:S3

direct product, metabelian, supersoluble, monomial

Aliases: C3xD6:S3, C33:4D4, C6.27S32, (S3xC6):1S3, (S3xC6):1C6, D6:1(C3xS3), C6.3(S3xC6), C3:Dic3:5C6, (C3xC6).40D6, C32:4(C3xD4), C32:11(C3:D4), (C32xC6).3C22, (S3xC3xC6):1C2, C2.3(C3xS32), C3:2(C3xC3:D4), (C3xC6).8(C2xC6), (C3xC3:Dic3):7C2, SmallGroup(216,121)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xD6:S3
C1C3C32C3xC6C32xC6S3xC3xC6 — C3xD6:S3
C32C3xC6 — C3xD6:S3
C1C6

Generators and relations for C3xD6: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, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2xC6, C3xS3, C3xC6, C3xC6, C3xC6, C3:D4, C3xD4, C33, C3xDic3, C3:Dic3, S3xC6, S3xC6, C62, S3xC32, C32xC6, D6:S3, C3xC3:D4, C3xC3:Dic3, S3xC3xC6, C3xD6:S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, C3xS3, C3:D4, C3xD4, S32, S3xC6, D6:S3, C3xC3:D4, C3xS32, C3xD6:S3

Permutation representations of C3xD6: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);

C3xD6:S3 is a maximal subgroup of
C33:D8  C33:7SD16  D6:4S32  (S3xC6):D6  (S3xC6).D6  D6.4S32  D6:S3:S3  C3xS3xC3: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+++++++-
imageC1C2C2C3C6C6S3D4D6C3xS3C3:D4C3xD4S3xC6C3xC3:D4S32D6:S3C3xS32C3xD6:S3
kernelC3xD6:S3C3xC3:Dic3S3xC3xC6D6:S3C3:Dic3S3xC6S3xC6C33C3xC6D6C32C32C6C3C6C3C2C1
# reps112224212442481122

Matrix representation of C3xD6: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] >;

C3xD6: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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