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G = C3xD6.4D6order 432 = 24·33

Direct product of C3 and D6.4D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xD6.4D6, C62.88D6, D6.4(S3xC6), D6:S3:4C6, (S3xDic3):2C6, (S3xC6).24D6, C32:2Q8:5C6, C33:22(C4oD4), C62.24(C2xC6), Dic3.4(S3xC6), (C3xDic3).27D6, (C3xC62).18C22, (C32xC6).31C23, C32:25(D4:2S3), (C32xDic3).12C22, C2.13(S32xC6), (C2xC6).46S32, C6.12(S3xC2xC6), C6.115(C2xS32), (C3xC3:D4):5S3, C3:D4:1(C3xS3), (C3xC3:D4):2C6, C22.2(C3xS32), (C3xS3xDic3):7C2, C3:4(C3xD4:2S3), (S3xC6).4(C2xC6), (C2xC6).15(S3xC6), C32:9(C3xC4oD4), (C2xC3:Dic3):11C6, (C6xC3:Dic3):15C2, (C32xC3:D4):2C2, (S3xC3xC6).13C22, (C3xD6:S3):11C2, (C3xC32:2Q8):11C2, C3:Dic3.19(C2xC6), (C3xC6).22(C22xC6), (C3xDic3).5(C2xC6), (C3xC6).136(C22xS3), (C3xC3:Dic3).56C22, SmallGroup(432,653)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xD6.4D6
C1C3C32C3xC6C32xC6S3xC3xC6C3xS3xDic3 — C3xD6.4D6
C32C3xC6 — C3xD6.4D6
C1C6C2xC6

Generators and relations for C3xD6.4D6
 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, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C4oD4, C3xS3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3:D4, C2xC12, C3xD4, C3xQ8, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C62, C62, C62, D4:2S3, C3xC4oD4, S3xC32, C32xC6, C32xC6, S3xDic3, D6:S3, C32:2Q8, C3xDic6, S3xC12, C6xDic3, C3xC3:D4, C3xC3:D4, C2xC3:Dic3, D4xC32, C32xDic3, C3xC3:Dic3, S3xC3xC6, C3xC62, D6.4D6, C3xD4:2S3, C3xS3xDic3, C3xD6:S3, C3xC32:2Q8, C32xC3:D4, C6xC3:Dic3, C3xD6.4D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C4oD4, C3xS3, C22xS3, C22xC6, S32, S3xC6, D4:2S3, C3xC4oD4, C2xS32, S3xC2xC6, C3xS32, D6.4D6, C3xD4:2S3, S32xC6, C3xD6.4D6

Permutation representations of C3xD6.4D6
On 24 points - transitive group 24T1283
Generators in S24
(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 16 7 22)(2 21 8 15)(3 14 9 20)(4 19 10 13)(5 24 11 18)(6 17 12 23)

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

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

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

G:=TransitiveGroup(24,1283);

72 conjugacy classes

class 1 2A2B2C2D3A3B3C···3H3I3J3K4A4B4C4D4E6A6B6C···6J6K···6Y6Z6AA6AB6AC6AD···6AI12A12B12C12D12E12F12G12H12I···12N12O12P
order12222333···333344444666···66···666666···6121212121212121212···121212
size11266112···2444669918112···24···4666612···126666999912···121818

72 irreducible representations

dim111111111111222222222244444444
type+++++++++++-+-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C4oD4C3xS3S3xC6S3xC6S3xC6C3xC4oD4S32D4:2S3C2xS32C3xS32D6.4D6C3xD4:2S3S32xC6C3xD6.4D6
kernelC3xD6.4D6C3xS3xDic3C3xD6:S3C3xC32:2Q8C32xC3:D4C6xC3:Dic3D6.4D6S3xDic3D6:S3C32:2Q8C3xC3:D4C2xC3:Dic3C3xC3:D4C3xDic3S3xC6C62C33C3:D4Dic3D6C2xC6C32C2xC6C32C6C22C3C3C2C1
# reps121121242242222224444412122424

Matrix representation of C3xD6.4D6 in GL4(F7) generated by

4000
0400
0040
0004
,
3656
4413
1135
1636
,
5122
6012
0010
6141
,
2313
2514
1423
1115
,
3026
1321
3102
2261
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] >;

C3xD6.4D6 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

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