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G = C3×C4.Dic3order 144 = 24·32

Direct product of C3 and C4.Dic3

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C4.Dic3, C12.1C12, C12.67D6, C62.5C4, C12.9Dic3, C327M4(2), C3⋊C85C6, C4.(C3×Dic3), C4.15(S3×C6), (C2×C6).6C12, C6.6(C2×C12), (C3×C12).5C4, (C6×C12).7C2, (C2×C12).6C6, (C2×C12).18S3, C12.16(C2×C6), C32(C3×M4(2)), (C2×C6).2Dic3, C2.3(C6×Dic3), C22.(C3×Dic3), C6.18(C2×Dic3), (C3×C12).45C22, (C3×C3⋊C8)⋊12C2, (C2×C4).2(C3×S3), (C3×C6).28(C2×C4), SmallGroup(144,75)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C4.Dic3
C1C3C6C12C3×C12C3×C3⋊C8 — C3×C4.Dic3
C3C6 — C3×C4.Dic3
C1C12C2×C12

Generators and relations for C3×C4.Dic3
 G = < a,b,c,d | a3=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

2C2
2C3
2C6
2C6
2C6
2C6
2C6
3C8
3C8
2C12
2C12
2C2×C6
2C3×C6
3M4(2)
2C2×C12
3C24
3C24
3C3×M4(2)

Permutation representations of C3×C4.Dic3
On 24 points - transitive group 24T211
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 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 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 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)

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

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

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

G:=TransitiveGroup(24,211);

C3×C4.Dic3 is a maximal subgroup of
C12.D12  C12.70D12  C12.14D12  C12.71D12  D124Dic3  C12.80D12  C12.82D12  C62.5Q8  D12.Dic3  C3⋊C8.22D6  C3⋊C820D6  D1218D6  D12.28D6  D12.29D6  Dic6.29D6  C3×S3×M4(2)  He37M4(2)  C36.C12  He38M4(2)
C3×C4.Dic3 is a maximal quotient of
He37M4(2)  C36.C12

54 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C···6M8A8B8C8D12A12B12C12D12E···12R24A···24H
order12233333444666···688881212121212···1224···24
size11211222112112···2666611112···26···6

54 irreducible representations

dim1111111111222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6Dic3M4(2)C3×S3C3×Dic3S3×C6C3×Dic3C4.Dic3C3×M4(2)C3×C4.Dic3
kernelC3×C4.Dic3C3×C3⋊C8C6×C12C4.Dic3C3×C12C62C3⋊C8C2×C12C12C2×C6C2×C12C12C12C2×C6C32C2×C4C4C4C22C3C3C1
# reps1212224244111122222448

Matrix representation of C3×C4.Dic3 in GL2(𝔽13) generated by

30
03
,
80
05
,
20
06
,
05
10
G:=sub<GL(2,GF(13))| [3,0,0,3],[8,0,0,5],[2,0,0,6],[0,1,5,0] >;

C3×C4.Dic3 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C3xC4.Dic3");
// GroupNames label

G:=SmallGroup(144,75);
// by ID

G=gap.SmallGroup(144,75);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,313,69,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

Export

Subgroup lattice of C3×C4.Dic3 in TeX

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