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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

Aliases: C22×C3≀C3, C32.1C62, C62.19C32, He33(C2×C6), (C2×He3)⋊2C6, (C3×C62)⋊2C3, (C32×C6)⋊5C6, C339(C2×C6), C6.8(C2×He3), (C2×C6).14He3, (C22×He3)⋊3C3, C3.2(C22×He3), (C2×3- 1+2)⋊1C6, 3- 1+21(C2×C6), (C22×3- 1+2)⋊3C3, (C3×C6).6(C3×C6), SmallGroup(324,86)

Series: Derived Chief Lower central Upper central

C1C32 — C22×C3≀C3
C1C3C32C33C3≀C3C2×C3≀C3 — C22×C3≀C3
C1C3C32 — C22×C3≀C3
C1C2×C6C62 — 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 2A2B2C3A3B3C···3J3K3L6A···6F6G···6AD6AE···6AJ9A9B9C9D18A···18L
order1222333···3336···66···66···6999918···18
size1111113···3991···13···39···999999···9

68 irreducible representations

dim111111113333
type++
imageC1C2C3C3C3C6C6C6He3C2×He3C3≀C3C2×C3≀C3
kernelC22×C3≀C3C2×C3≀C3C22×He3C22×3- 1+2C3×C62C2×He3C2×3- 1+2C32×C6C2×C6C6C22C2
# reps13242612626618

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

180000
018000
00100
00010
00001
,
10000
018000
00100
00010
00001
,
10000
01000
001100
00070
00001
,
10000
01000
00700
00070
00007
,
70000
07000
00001
001800
000180
,
70000
07000
00700
00010
00001

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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