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G = C3×D6.4D6order 432 = 24·33

Direct product of C3 and D6.4D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×D6.4D6, C62.88D6, D6.4(S3×C6), D6⋊S34C6, (S3×Dic3)⋊2C6, (S3×C6).24D6, C322Q85C6, C3322(C4○D4), C62.24(C2×C6), Dic3.4(S3×C6), (C3×Dic3).27D6, (C3×C62).18C22, (C32×C6).31C23, C3225(D42S3), (C32×Dic3).12C22, C2.13(S32×C6), (C2×C6).46S32, C6.12(S3×C2×C6), C6.115(C2×S32), (C3×C3⋊D4)⋊5S3, C3⋊D41(C3×S3), (C3×C3⋊D4)⋊2C6, C22.2(C3×S32), (C3×S3×Dic3)⋊7C2, C34(C3×D42S3), (S3×C6).4(C2×C6), (C2×C6).15(S3×C6), C329(C3×C4○D4), (C2×C3⋊Dic3)⋊11C6, (C6×C3⋊Dic3)⋊15C2, (C32×C3⋊D4)⋊2C2, (S3×C3×C6).13C22, (C3×D6⋊S3)⋊11C2, (C3×C322Q8)⋊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

C1C3×C6 — C3×D6.4D6
C1C3C32C3×C6C32×C6S3×C3×C6C3×S3×Dic3 — C3×D6.4D6
C32C3×C6 — C3×D6.4D6
C1C6C2×C6

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], D42S3 [×2], C3×C4○D4, S3×C32 [×2], C32×C6, C32×C6, S3×Dic3 [×2], D6⋊S3, C322Q8, 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×D42S3 [×2], C3×S3×Dic3 [×2], C3×D6⋊S3, C3×C322Q8, 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], D42S3 [×2], C3×C4○D4, C2×S32, S3×C2×C6 [×2], C3×S32, D6.4D6, C3×D42S3 [×2], S32×C6, C3×D6.4D6

Permutation representations of C3×D6.4D6
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 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 2A2B2C2D3A3B3C···3H3I3J3K4A4B4C4D4E6A6B6C···6J6K···6Y6Z6AA6AB6AC6AD···6AI12A12B12C12D12E12F12G12H12I···12N12O12P
order12222333···333344444666···66···666666···6121212121212121212···121212
size11266112···2444669918112···24···4666612···126666999912···121818

72 irreducible representations

dim111111111111222222222244444444
type+++++++++++-+-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C4○D4C3×S3S3×C6S3×C6S3×C6C3×C4○D4S32D42S3C2×S32C3×S32D6.4D6C3×D42S3S32×C6C3×D6.4D6
kernelC3×D6.4D6C3×S3×Dic3C3×D6⋊S3C3×C322Q8C32×C3⋊D4C6×C3⋊Dic3D6.4D6S3×Dic3D6⋊S3C322Q8C3×C3⋊D4C2×C3⋊Dic3C3×C3⋊D4C3×Dic3S3×C6C62C33C3⋊D4Dic3D6C2×C6C32C2×C6C32C6C22C3C3C2C1
# reps121121242242222224444412122424

Matrix representation of C3×D6.4D6 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] >;

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

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