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## G = C3×C2≀C22order 192 = 26·3

### Direct product of C3 and C2≀C22

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C2≀C22
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2 [×8], C3, C4 [×6], C22 [×3], C22 [×18], C6, C6 [×8], C2×C4 [×3], C2×C4 [×6], D4 [×15], Q8, C23, C23 [×3], C23 [×6], C12 [×6], C2×C6 [×3], C2×C6 [×18], C22⋊C4 [×3], C22⋊C4 [×3], C2×D4 [×3], C2×D4 [×6], C4○D4 [×3], C24, C2×C12 [×3], C2×C12 [×6], C3×D4 [×15], C3×Q8, C22×C6, C22×C6 [×3], C22×C6 [×6], C23⋊C4 [×3], C22≀C2 [×3], 2+ 1+4, C3×C22⋊C4 [×3], C3×C22⋊C4 [×3], C6×D4 [×3], C6×D4 [×6], C3×C4○D4 [×3], C23×C6, C2≀C22, C3×C23⋊C4 [×3], C3×C22≀C2 [×3], C3×2+ 1+4, C3×C2≀C22
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, C2×C6 [×7], C2×D4 [×3], C3×D4 [×6], C22×C6, C22≀C2, C6×D4 [×3], C2≀C22, C3×C22≀C2, C3×C2≀C22

Permutation representations of C3×C2≀C22
On 24 points - transitive group 24T286
Generators in S24
(1 10 8)(2 9 7)(3 12 6)(4 11 5)(13 20 23)(14 17 24)(15 18 21)(16 19 22)
(1 17)(2 20)(3 18)(4 19)(5 16)(6 15)(7 13)(8 14)(9 23)(10 24)(11 22)(12 21)
(2 3)(6 7)(9 12)(13 15)(18 20)(21 23)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(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)
(14 16)(17 19)(22 24)

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

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

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

G:=TransitiveGroup(24,286);

On 24 points - transitive group 24T352
Generators in S24
(1 15 21)(2 16 22)(3 13 23)(4 14 24)(5 9 20)(6 10 17)(7 11 18)(8 12 19)
(1 5)(2 8)(3 4)(6 7)(9 15)(10 11)(12 16)(13 14)(17 18)(19 22)(20 21)(23 24)
(1 3)(2 7)(4 5)(6 8)(9 14)(10 12)(11 16)(13 15)(17 19)(18 22)(20 24)(21 23)
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 23)(18 22)(19 21)(20 24)

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

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

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

G:=TransitiveGroup(24,352);

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 C3×D4 C3×D4 C2≀C22 C3×C2≀C22 kernel C3×C2≀C22 C3×C23⋊C4 C3×C22≀C2 C3×2+ 1+4 C2≀C22 C23⋊C4 C22≀C2 2+ 1+4 C2×C12 C22×C6 C2×C4 C23 C3 C1 # reps 1 3 3 1 2 6 6 2 3 3 6 6 2 4

Matrix representation of C3×C2≀C22 in GL4(𝔽7) generated by

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

C3×C2≀C22 in GAP, Magma, Sage, TeX

C_3\times C_2\wr C_2^2
% in TeX

G:=Group("C3xC2wrC2^2");
// GroupNames label

G:=SmallGroup(192,890);
// by ID

G=gap.SmallGroup(192,890);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,1068,3036]);
// Polycyclic

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

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