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## G = C3×C2≀C4order 192 = 26·3

### Direct product of C3 and C2≀C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×C2≀C4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C6×D4 — C3×C23⋊C4 — C3×C2≀C4
 Lower central C1 — C2 — C22 — C23 — C3×C2≀C4
 Upper central C1 — C6 — C2×C6 — C6×D4 — C3×C2≀C4

Generators and relations for C3×C2≀C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 258 in 94 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], C6, C6 [×5], C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], C12 [×3], C2×C6, C2×C6 [×12], C22⋊C4, C22⋊C4 [×2], M4(2), C2×D4, C2×D4, C24, C24, C2×C12, C2×C12 [×2], C3×D4 [×3], C22×C6 [×2], C22×C6 [×4], C23⋊C4, C4.D4, C22≀C2, C3×C22⋊C4, C3×C22⋊C4 [×2], C3×M4(2), C6×D4, C6×D4, C23×C6, C2≀C4, C3×C23⋊C4, C3×C4.D4, C3×C22≀C2, C3×C2≀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, C2≀C4, C3×C23⋊C4, C3×C2≀C4

Permutation representations of C3×C2≀C4
On 24 points - transitive group 24T285
Generators in S24
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
(2 12)(8 15)(20 24)
(2 12)(3 9)(5 16)(8 15)(17 21)(20 24)
(2 12)(4 10)(6 13)(8 15)(18 22)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 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)

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

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

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

G:=TransitiveGroup(24,285);

On 24 points - transitive group 24T351
Generators in S24
(1 12 7)(2 11 8)(3 5 10)(4 6 9)(13 24 20)(14 21 17)(15 22 18)(16 23 19)
(1 15)(3 20)(5 13)(7 18)(10 24)(12 22)
(1 15)(2 16)(3 20)(4 17)(5 13)(6 14)(7 18)(8 19)(9 21)(10 24)(11 23)(12 22)
(1 5)(3 7)(10 12)(13 15)(18 20)(22 24)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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)

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

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

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

G:=TransitiveGroup(24,351);

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E ··· 6L 8A 8B 12A 12B 12C ··· 12H 24A 24B 24C 24D order 1 2 2 2 2 2 2 3 3 4 4 4 4 6 6 6 6 6 ··· 6 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 2 4 4 4 4 1 1 4 8 8 8 1 1 2 2 4 ··· 4 8 8 4 4 8 ··· 8 8 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 C3×D4 C3×D4 C23⋊C4 C2≀C4 C3×C23⋊C4 C3×C2≀C4 kernel C3×C2≀C4 C3×C23⋊C4 C3×C4.D4 C3×C22≀C2 C2≀C4 C3×C22⋊C4 C23×C6 C23⋊C4 C4.D4 C22≀C2 C22⋊C4 C24 C2×C12 C22×C6 C2×C4 C23 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 1 2 2 4

Matrix representation of C3×C2≀C4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 1 0 4 0 0 1 5 0 0 0 6 0 0 0 0 1
,
 1 0 4 0 1 0 1 5 0 0 6 0 4 3 5 0
,
 0 6 3 2 6 0 4 2 0 0 6 0 0 0 0 1
,
 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 2 0 4 1 0 0 2 1 2 5 6 4 4 4 5 6
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,4,5,6,0,0,0,0,1],[1,1,0,4,0,0,0,3,4,1,6,5,0,5,0,0],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,0,2,4,0,0,5,4,4,2,6,5,1,1,4,6] >;

C3×C2≀C4 in GAP, Magma, Sage, TeX

C_3\times C_2\wr C_4
% in TeX

G:=Group("C3xC2wrC4");
// GroupNames label

G:=SmallGroup(192,157);
// by ID

G=gap.SmallGroup(192,157);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1271,375,6053]);
// Polycyclic

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

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