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G = C3×C4.Dic3order 144 = 24·32

Direct product of C3 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8 — C3×C4.Dic3
 Lower central C3 — C6 — C3×C4.Dic3
 Upper central C1 — C12 — C2×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 >

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 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6M 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12R 24A ··· 24H order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 1 1 2 2 2 1 1 2 1 1 2 ··· 2 6 6 6 6 1 1 1 1 2 ··· 2 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) C3×C4.Dic3 kernel C3×C4.Dic3 C3×C3⋊C8 C6×C12 C4.Dic3 C3×C12 C62 C3⋊C8 C2×C12 C12 C2×C6 C2×C12 C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 1 1 1 2 2 2 2 2 4 4 8

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

 3 0 0 3
,
 8 0 0 5
,
 2 0 0 6
,
 0 5 1 0
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

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