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G = C6×C3≀C3  order 486 = 2·35

Direct product of C6 and C3≀C3

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

Aliases: C6×C3≀C3, C34⋊12C6, He3⋊2(C3×C6), (C6×He3)⋊3C3, (C33×C6)⋊2C3, C6.7(C3×He3), C3.7(C6×He3), (C3×He3)⋊15C6, C33⋊11(C3×C6), (C3×C6).27He3, (C3×C6).1C33, (C2×He3)⋊1C32, (C32×C6)⋊4C32, C32.25(C2×He3), C32.1(C32×C6), (C6×3- 1+2)⋊5C3, 3- 1+2⋊1(C3×C6), (C3×3- 1+2)⋊12C6, (C2×3- 1+2)⋊1C32, SmallGroup(486,210)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C6×C3≀C3
Chief series
C1 — C3 — C32 — C33 — C34 — C3×C3≀C3 — C6×C3≀C3
Lower central
C1 — C3 — C32 — C6×C3≀C3
Upper central
C1 — C3×C6 — C32×C6 — C6×C3≀C3

Generators and relations for C6×C3≀C3
 G = < a,b,c,d,e | a6=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

Subgroups: 684 in 252 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C33, C3×C18, C2×He3, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C32×C6, C32×C6, C3≀C3, C3×He3, C3×3- 1+2, C34, C2×C3≀C3, C6×He3, C6×3- 1+2, C33×C6, C3×C3≀C3, C6×C3≀C3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3≀C3, C3×He3, C2×C3≀C3, C6×He3, C3×C3≀C3, C6×C3≀C3

Smallest permutation representation of C6×C3≀C3
►On 54 points
Generators in S54
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)

(1 3 5)(2 4 6)(7 38 30)(8 39 25)(9 40 26)(10 41 27)(11 42 28)(12 37 29)(13 15 17)(14 16 18)(19 34 43)(20 35 44)(21 36 45)(22 31 46)(23 32 47)(24 33 48)(49 51 53)(50 52 54)

(1 49 15)(2 50 16)(3 51 17)(4 52 18)(5 53 13)(6 54 14)(7 42 26)(8 37 27)(9 38 28)(10 39 29)(11 40 30)(12 41 25)(19 45 32)(20 46 33)(21 47 34)(22 48 35)(23 43 36)(24 44 31)

(1 34 12)(2 35 7)(3 36 8)(4 31 9)(5 32 10)(6 33 11)(13 45 29)(14 46 30)(15 47 25)(16 48 26)(17 43 27)(18 44 28)(19 39 53)(20 40 54)(21 41 49)(22 42 50)(23 37 51)(24 38 52)

(1 5 3)(2 6 4)(13 17 15)(14 18 16)(19 47 36)(20 48 31)(21 43 32)(22 44 33)(23 45 34)(24 46 35)(49 53 51)(50 54 52)
 
G:=sub<Sym(54)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54), (1,3,5)(2,4,6)(7,38,30)(8,39,25)(9,40,26)(10,41,27)(11,42,28)(12,37,29)(13,15,17)(14,16,18)(19,34,43)(20,35,44)(21,36,45)(22,31,46)(23,32,47)(24,33,48)(49,51,53)(50,52,54), (1,49,15)(2,50,16)(3,51,17)(4,52,18)(5,53,13)(6,54,14)(7,42,26)(8,37,27)(9,38,28)(10,39,29)(11,40,30)(12,41,25)(19,45,32)(20,46,33)(21,47,34)(22,48,35)(23,43,36)(24,44,31), (1,34,12)(2,35,7)(3,36,8)(4,31,9)(5,32,10)(6,33,11)(13,45,29)(14,46,30)(15,47,25)(16,48,26)(17,43,27)(18,44,28)(19,39,53)(20,40,54)(21,41,49)(22,42,50)(23,37,51)(24,38,52), (1,5,3)(2,6,4)(13,17,15)(14,18,16)(19,47,36)(20,48,31)(21,43,32)(22,44,33)(23,45,34)(24,46,35)(49,53,51)(50,54,52)>;
 
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)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54), (1,3,5)(2,4,6)(7,38,30)(8,39,25)(9,40,26)(10,41,27)(11,42,28)(12,37,29)(13,15,17)(14,16,18)(19,34,43)(20,35,44)(21,36,45)(22,31,46)(23,32,47)(24,33,48)(49,51,53)(50,52,54), (1,49,15)(2,50,16)(3,51,17)(4,52,18)(5,53,13)(6,54,14)(7,42,26)(8,37,27)(9,38,28)(10,39,29)(11,40,30)(12,41,25)(19,45,32)(20,46,33)(21,47,34)(22,48,35)(23,43,36)(24,44,31), (1,34,12)(2,35,7)(3,36,8)(4,31,9)(5,32,10)(6,33,11)(13,45,29)(14,46,30)(15,47,25)(16,48,26)(17,43,27)(18,44,28)(19,39,53)(20,40,54)(21,41,49)(22,42,50)(23,37,51)(24,38,52), (1,5,3)(2,6,4)(13,17,15)(14,18,16)(19,47,36)(20,48,31)(21,43,32)(22,44,33)(23,45,34)(24,46,35)(49,53,51)(50,54,52) );
 
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),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54)], [(1,3,5),(2,4,6),(7,38,30),(8,39,25),(9,40,26),(10,41,27),(11,42,28),(12,37,29),(13,15,17),(14,16,18),(19,34,43),(20,35,44),(21,36,45),(22,31,46),(23,32,47),(24,33,48),(49,51,53),(50,52,54)], [(1,49,15),(2,50,16),(3,51,17),(4,52,18),(5,53,13),(6,54,14),(7,42,26),(8,37,27),(9,38,28),(10,39,29),(11,40,30),(12,41,25),(19,45,32),(20,46,33),(21,47,34),(22,48,35),(23,43,36),(24,44,31)], [(1,34,12),(2,35,7),(3,36,8),(4,31,9),(5,32,10),(6,33,11),(13,45,29),(14,46,30),(15,47,25),(16,48,26),(17,43,27),(18,44,28),(19,39,53),(20,40,54),(21,41,49),(22,42,50),(23,37,51),(24,38,52)], [(1,5,3),(2,6,4),(13,17,15),(14,18,16),(19,47,36),(20,48,31),(21,43,32),(22,44,33),(23,45,34),(24,46,35),(49,53,51),(50,54,52)]])
 

102 conjugacy classes

class 1  2 3A···3H3I···3AF3AG···3AL6A···6H6I···6AF6AG···6AL9A···9L18A···18L
order123···33···33···36···66···66···69···918···18
size111···13···39···91···13···39···99···99···9

102 irreducible representations

dim11111111113333
type++
imageC1C2C3C3C3C3C6C6C6C6He3C2×He3C3≀C3C2×C3≀C3
kernelC6×C3≀C3C3×C3≀C3C2×C3≀C3C6×He3C6×3- 1+2C33×C6C3≀C3C3×He3C3×3- 1+2C34C3×C6C32C6C3
# reps111824218242661818

Matrix representation of C6×C3≀C3 ►in GL4(𝔽19) generated by

8000
01100
00110
00011
,
1000
0700
00110
0001
,
1000
0700
0070
0007
,
1000
00110
00011
01100
,
1000
01100
0010
00011
G:=sub<GL(4,GF(19))| [8,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,7,0,0,0,0,11,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,0,0,11,0,11,0,0,0,0,11,0],[1,0,0,0,0,11,0,0,0,0,1,0,0,0,0,11] >;
 

C6×C3≀C3 in GAP, Magma, Sage, TeX 

C_6\times C_3\wr C_3
 
% in TeX
 
G:=Group("C6xC3wrC3");
 
// GroupNames label
 
G:=SmallGroup(486,210);
 
// by ID
 
G=gap.SmallGroup(486,210);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c^-1*d>;
 
// generators/relations
 

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