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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, C3, C3, C22, C6, C6, C9, C32, C32, C2×C6, C2×C6, C18, C3×C6, C3×C6, He3, 3- 1+2, C33, C2×C18, C62, C62, C2×He3, C2×3- 1+2, C32×C6, C3≀C3, C22×He3, C22×3- 1+2, C3×C62, C2×C3≀C3, C22×C3≀C3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, He3, C62, C2×He3, C3≀C3, C22×He3, C2×C3≀C3, C22×C3≀C3

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