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G = C3×C4≀C2order 96 = 25·3

Direct product of C3 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×C4≀C2, D42C12, C426C6, Q83C12, C12.66D4, M4(2)⋊4C6, (C3×D4)⋊5C4, (C3×Q8)⋊5C4, (C4×C12)⋊10C2, C4.3(C2×C12), C4○D4.3C6, (C2×C6).22D4, C4.17(C3×D4), C12.30(C2×C4), C22.3(C3×D4), C6.26(C22⋊C4), (C3×M4(2))⋊10C2, (C2×C12).116C22, (C2×C4).19(C2×C6), (C3×C4○D4).4C2, C2.8(C3×C22⋊C4), SmallGroup(96,54)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C4≀C2
C1C2C4C2×C4C2×C12C3×M4(2) — C3×C4≀C2
C1C2C4 — C3×C4≀C2
C1C12C2×C12 — C3×C4≀C2

Generators and relations for C3×C4≀C2
 G = < a,b,c,d | a3=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

2C2
4C2
2C4
2C22
2C4
2C4
2C6
4C6
2D4
2C2×C4
2C2×C4
2C8
2C2×C6
2C12
2C12
2C12
2C2×C12
2C24
2C3×D4
2C2×C12

Permutation representations of C3×C4≀C2
On 24 points - transitive group 24T113
Generators in S24
(1 11 7)(2 12 8)(3 9 5)(4 10 6)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(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)(2 15)(3 14)(4 13)(5 18)(6 17)(7 20)(8 19)(9 22)(10 21)(11 24)(12 23)
(13 16 15 14)(17 20 19 18)(21 24 23 22)

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

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

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

G:=TransitiveGroup(24,113);

C3×C4≀C2 is a maximal subgroup of   C423D6  Q85D12  M4(2).22D6  C42.196D6  C425D6  Q8.14D12  D4.10D12

42 conjugacy classes

class 1 2A2B2C3A3B4A4B4C···4G4H6A6B6C6D6E6F8A8B12A12B12C12D12E···12N12O12P24A24B24C24D
order122233444···44666666881212121212···12121224242424
size112411112···241122444411112···2444444

42 irreducible representations

dim111111111111222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12D4D4C3×D4C3×D4C4≀C2C3×C4≀C2
kernelC3×C4≀C2C4×C12C3×M4(2)C3×C4○D4C4≀C2C3×D4C3×Q8C42M4(2)C4○D4D4Q8C12C2×C6C4C22C3C1
# reps111122222244112248

Matrix representation of C3×C4≀C2 in GL2(𝔽13) generated by

90
09
,
02
60
,
011
60
,
114
1211
G:=sub<GL(2,GF(13))| [9,0,0,9],[0,6,2,0],[0,6,11,0],[11,12,4,11] >;

C3×C4≀C2 in GAP, Magma, Sage, TeX

C_3\times C_4\wr C_2
% in TeX

G:=Group("C3xC4wrC2");
// GroupNames label

G:=SmallGroup(96,54);
// by ID

G=gap.SmallGroup(96,54);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,729,117,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4≀C2 in TeX

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