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## G = C3×C4○D12order 144 = 24·32

### Direct product of C3 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C4○D12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12 — C3×C4○D12
 Lower central C3 — C6 — C3×C4○D12
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4○D12
G = < a,b,c,d | a3=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 168 in 88 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C3×C4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, C4○D12, C3×C4○D4, S3×C2×C6, C3×C4○D12

Permutation representations of C3×C4○D12
On 24 points - transitive group 24T210
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)

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

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

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

G:=TransitiveGroup(24,210);

C3×C4○D12 is a maximal subgroup of
D124Dic3  D122Dic3  D12.2Dic3  D12.Dic3  D12.30D6  D1220D6  D1218D6  D12.32D6  D12.27D6  D12.29D6  D12.33D6  D12.34D6  D1223D6  D1224D6  D1227D6  C3×S3×C4○D4  C62.36D6  D366C6  C62.47D6
C3×C4○D12 is a maximal quotient of
C12×Dic6  C12×D12  C12×C3⋊D4  C62.36D6  D366C6

54 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6M 6N 6O 6P 6Q 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 2 6 6 1 1 2 2 2 1 1 2 6 6 1 1 2 ··· 2 6 6 6 6 1 1 1 1 2 ··· 2 6 6 6 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 kernel C3×C4○D12 C3×Dic6 S3×C12 C3×D12 C3×C3⋊D4 C6×C12 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C2×C12 C12 C2×C6 C32 C2×C4 C4 C22 C3 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 1 2 1 2 2 4 2 4 4 8

Matrix representation of C3×C4○D12 in GL2(𝔽13) generated by

 3 0 0 3
,
 5 0 0 5
,
 6 11 11 3
,
 10 4 11 3
G:=sub<GL(2,GF(13))| [3,0,0,3],[5,0,0,5],[6,11,11,3],[10,11,4,3] >;

C3×C4○D12 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{12}
% in TeX

G:=Group("C3xC4oD12");
// GroupNames label

G:=SmallGroup(144,161);
// by ID

G=gap.SmallGroup(144,161);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,3461]);
// Polycyclic

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

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