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## G = C6×C3≀C3order 486 = 2·35

### Direct product of C6 and C3≀C3

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

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 ··· 3H 3I ··· 3AF 3AG ··· 3AL 6A ··· 6H 6I ··· 6AF 6AG ··· 6AL 9A ··· 9L 18A ··· 18L order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9 9 ··· 9 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 He3 C2×He3 C3≀C3 C2×C3≀C3 kernel C6×C3≀C3 C3×C3≀C3 C2×C3≀C3 C6×He3 C6×3- 1+2 C33×C6 C3≀C3 C3×He3 C3×3- 1+2 C34 C3×C6 C32 C6 C3 # reps 1 1 18 2 4 2 18 2 4 2 6 6 18 18

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

 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 11 0 0 0 0 11 0 11 0 0
,
 1 0 0 0 0 11 0 0 0 0 1 0 0 0 0 11
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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