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## G = C22×C3≀C3order 324 = 22·34

### Direct product of C22 and C3≀C3

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C22×C3≀C3
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C2×C3≀C3 — C22×C3≀C3
 Lower central C1 — C3 — C32 — C22×C3≀C3
 Upper central C1 — C2×C6 — C62 — C22×C3≀C3

Generators and relations for C22×C3≀C3
G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, cf=fc, de=ed, df=fd, fef-1=cd-1e >

Subgroups: 250 in 100 conjugacy classes, 40 normal (12 characteristic)
C1, C2 [×3], C3, C3 [×5], C22, C6 [×3], C6 [×15], C9 [×2], C32, C32 [×5], C2×C6, C2×C6 [×5], C18 [×6], C3×C6 [×3], C3×C6 [×15], He3, 3- 1+2 [×2], C33, C2×C18 [×2], C62, C62 [×5], C2×He3 [×3], C2×3- 1+2 [×6], C32×C6 [×3], C3≀C3, C22×He3, C22×3- 1+2 [×2], C3×C62, C2×C3≀C3 [×3], C22×C3≀C3
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, C2×C6 [×4], C3×C6 [×3], He3, C62, C2×He3 [×3], C3≀C3, C22×He3, C2×C3≀C3 [×3], C22×C3≀C3

Smallest permutation representation of C22×C3≀C3
On 36 points
Generators in S36
(1 8)(2 3)(4 11)(5 7)(6 10)(9 12)(13 35)(14 36)(15 34)(16 23)(17 24)(18 22)(19 31)(20 32)(21 33)(25 29)(26 30)(27 28)
(1 6)(2 11)(3 4)(5 9)(7 12)(8 10)(13 33)(14 31)(15 32)(16 25)(17 26)(18 27)(19 36)(20 34)(21 35)(22 28)(23 29)(24 30)
(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 5 3)(2 8 7)(4 6 9)(10 12 11)(13 15 14)(16 17 18)(19 21 20)(22 23 24)(25 26 27)(28 29 30)(31 33 32)(34 36 35)
(1 14 17)(2 34 23)(3 15 16)(4 32 25)(5 13 18)(6 31 26)(7 35 22)(8 36 24)(9 33 27)(10 19 30)(11 20 29)(12 21 28)
(1 5 3)(2 8 7)(4 6 9)(10 12 11)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)

G:=sub<Sym(36)| (1,8)(2,3)(4,11)(5,7)(6,10)(9,12)(13,35)(14,36)(15,34)(16,23)(17,24)(18,22)(19,31)(20,32)(21,33)(25,29)(26,30)(27,28), (1,6)(2,11)(3,4)(5,9)(7,12)(8,10)(13,33)(14,31)(15,32)(16,25)(17,26)(18,27)(19,36)(20,34)(21,35)(22,28)(23,29)(24,30), (13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,5,3)(2,8,7)(4,6,9)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,26,27)(28,29,30)(31,33,32)(34,36,35), (1,14,17)(2,34,23)(3,15,16)(4,32,25)(5,13,18)(6,31,26)(7,35,22)(8,36,24)(9,33,27)(10,19,30)(11,20,29)(12,21,28), (1,5,3)(2,8,7)(4,6,9)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)>;

G:=Group( (1,8)(2,3)(4,11)(5,7)(6,10)(9,12)(13,35)(14,36)(15,34)(16,23)(17,24)(18,22)(19,31)(20,32)(21,33)(25,29)(26,30)(27,28), (1,6)(2,11)(3,4)(5,9)(7,12)(8,10)(13,33)(14,31)(15,32)(16,25)(17,26)(18,27)(19,36)(20,34)(21,35)(22,28)(23,29)(24,30), (13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,5,3)(2,8,7)(4,6,9)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,26,27)(28,29,30)(31,33,32)(34,36,35), (1,14,17)(2,34,23)(3,15,16)(4,32,25)(5,13,18)(6,31,26)(7,35,22)(8,36,24)(9,33,27)(10,19,30)(11,20,29)(12,21,28), (1,5,3)(2,8,7)(4,6,9)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36) );

G=PermutationGroup([(1,8),(2,3),(4,11),(5,7),(6,10),(9,12),(13,35),(14,36),(15,34),(16,23),(17,24),(18,22),(19,31),(20,32),(21,33),(25,29),(26,30),(27,28)], [(1,6),(2,11),(3,4),(5,9),(7,12),(8,10),(13,33),(14,31),(15,32),(16,25),(17,26),(18,27),(19,36),(20,34),(21,35),(22,28),(23,29),(24,30)], [(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,5,3),(2,8,7),(4,6,9),(10,12,11),(13,15,14),(16,17,18),(19,21,20),(22,23,24),(25,26,27),(28,29,30),(31,33,32),(34,36,35)], [(1,14,17),(2,34,23),(3,15,16),(4,32,25),(5,13,18),(6,31,26),(7,35,22),(8,36,24),(9,33,27),(10,19,30),(11,20,29),(12,21,28)], [(1,5,3),(2,8,7),(4,6,9),(10,12,11),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)])

68 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3J 3K 3L 6A ··· 6F 6G ··· 6AD 6AE ··· 6AJ 9A 9B 9C 9D 18A ··· 18L order 1 2 2 2 3 3 3 ··· 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 9 9 9 9 18 ··· 18 size 1 1 1 1 1 1 3 ··· 3 9 9 1 ··· 1 3 ··· 3 9 ··· 9 9 9 9 9 9 ··· 9

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 C2×He3 C3≀C3 C2×C3≀C3 kernel C22×C3≀C3 C2×C3≀C3 C22×He3 C22×3- 1+2 C3×C62 C2×He3 C2×3- 1+2 C32×C6 C2×C6 C6 C22 C2 # reps 1 3 2 4 2 6 12 6 2 6 6 18

Matrix representation of C22×C3≀C3 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 7 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 7 0 0 0 0 0 7 0 0 0 0 0 0 0 1 0 0 18 0 0 0 0 0 18 0
,
 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[7,0,0,0,0,0,7,0,0,0,0,0,0,18,0,0,0,0,0,18,0,0,1,0,0],[7,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1] >;

C22×C3≀C3 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC3wrC3");
// GroupNames label

G:=SmallGroup(324,86);
// by ID

G=gap.SmallGroup(324,86);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,1096]);
// Polycyclic

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

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