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G = C3×C42⋊C4order 192 = 26·3

Direct product of C3 and C42⋊C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C3×C42⋊C4, C424C12, (C4×C12)⋊5C4, (C6×D4)⋊4C4, (C2×D4)⋊2C12, C23⋊C42C6, C41D4.2C6, C23.3(C3×D4), (C22×C6).3D4, C6.34(C23⋊C4), (C6×D4).176C22, (C3×C23⋊C4)⋊8C2, (C2×C4).1(C2×C12), (C2×D4).3(C2×C6), C2.8(C3×C23⋊C4), (C2×C12).12(C2×C4), (C3×C41D4).9C2, (C2×C6).75(C22⋊C4), C22.12(C3×C22⋊C4), SmallGroup(192,159)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C3×C42⋊C4
C1C2C22C23C2×D4C6×D4C3×C23⋊C4 — C3×C42⋊C4
C1C2C22C2×C4 — C3×C42⋊C4
C1C6C2×C6C6×D4 — C3×C42⋊C4

Generators and relations for C3×C42⋊C4
 G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 242 in 86 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2 [×4], C3, C4 [×5], C22, C22 [×7], C6, C6 [×4], C2×C4, C2×C4 [×3], D4 [×6], C23 [×2], C23, C12 [×5], C2×C6, C2×C6 [×7], C42, C22⋊C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×C12, C2×C12 [×3], C3×D4 [×6], C22×C6 [×2], C22×C6, C23⋊C4 [×2], C41D4, C4×C12, C3×C22⋊C4 [×2], C6×D4 [×2], C6×D4 [×2], C42⋊C4, C3×C23⋊C4 [×2], C3×C41D4, C3×C42⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C12 [×2], C2×C6, C22⋊C4, C2×C12, C3×D4 [×2], C23⋊C4, C3×C22⋊C4, C42⋊C4, C3×C23⋊C4, C3×C42⋊C4

Permutation representations of C3×C42⋊C4
On 24 points - transitive group 24T353
Generators in S24
(1 5 9)(2 10 4)(3 8 7)(6 12 11)(13 22 18)(14 23 19)(15 24 20)(16 21 17)
(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 11 4)(2 5 8 6)(7 12 10 9)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 13 4 16)(2 21 5 22)(3 14 11 15)(6 24 8 23)(7 19 12 20)(9 18 10 17)

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

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

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

G:=TransitiveGroup(24,353);

39 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F4G6A6B6C6D6E6F6G6H6I6J12A···12F12G···12N
order122222334444444666666666612···1212···12
size11244811444888811224444884···48···8

39 irreducible representations

dim1111111111224444
type++++++
imageC1C2C2C3C4C4C6C6C12C12D4C3×D4C23⋊C4C42⋊C4C3×C23⋊C4C3×C42⋊C4
kernelC3×C42⋊C4C3×C23⋊C4C3×C41D4C42⋊C4C4×C12C6×D4C23⋊C4C41D4C42C2×D4C22×C6C23C6C3C2C1
# reps1212224244241224

Matrix representation of C3×C42⋊C4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
2301
1015
4446
0006
,
4546
6060
3331
3420
,
1614
6661
2563
3321
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,1,4,0,3,0,4,0,0,1,4,0,1,5,6,6],[4,6,3,3,5,0,3,4,4,6,3,2,6,0,1,0],[1,6,2,3,6,6,5,3,1,6,6,2,4,1,3,1] >;

C3×C42⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_4
% in TeX

G:=Group("C3xC4^2:C4");
// GroupNames label

G:=SmallGroup(192,159);
// by ID

G=gap.SmallGroup(192,159);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951,375,6053]);
// Polycyclic

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

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