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## G = C3×C22≀C2order 96 = 25·3

### Direct product of C3 and C22≀C2

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C22≀C2
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×6], C22 [×17], C6 [×3], C6 [×7], C2×C4 [×3], D4 [×6], C23, C23 [×3], C23 [×6], C12 [×3], C2×C6, C2×C6 [×6], C2×C6 [×17], C22⋊C4 [×3], C2×D4 [×3], C24, C2×C12 [×3], C3×D4 [×6], C22×C6, C22×C6 [×3], C22×C6 [×6], C22≀C2, C3×C22⋊C4 [×3], C6×D4 [×3], C23×C6, C3×C22≀C2
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], C3×C22≀C2

Permutation representations of C3×C22≀C2
On 24 points - transitive group 24T112
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)
(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(1 10)(2 11)(3 12)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)

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

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

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

G:=TransitiveGroup(24,112);

C3×C22≀C2 is a maximal subgroup of
C245Dic3  C246D6  C24.67D6  C24.43D6  C247D6  C248D6  C24.44D6  C24.45D6  C24.46D6  C249D6  C24.47D6  C3×D42  C24⋊C18

42 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 3A 3B 4A 4B 4C 6A ··· 6F 6G ··· 6R 6S 6T 12A ··· 12F order 1 2 2 2 2 ··· 2 2 3 3 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 size 1 1 1 1 2 ··· 2 4 1 1 4 4 4 1 ··· 1 2 ··· 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel C3×C22≀C2 C3×C22⋊C4 C6×D4 C23×C6 C22≀C2 C22⋊C4 C2×D4 C24 C2×C6 C22 # reps 1 3 3 1 2 6 6 2 6 12

Matrix representation of C3×C22≀C2 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 12 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C22≀C2 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC2^2wrC2");
// GroupNames label

G:=SmallGroup(96,167);
// by ID

G=gap.SmallGroup(96,167);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,938]);
// Polycyclic

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

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