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## G = C6×3- 1+2order 162 = 2·34

### Direct product of C6 and 3- 1+2

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C6×3- 1+2, C18⋊C32, C6.2C33, C33.4C6, C92(C3×C6), (C3×C9)⋊11C6, (C3×C18)⋊4C3, C3.2(C32×C6), (C3×C6).8C32, (C32×C6).2C3, C32.11(C3×C6), SmallGroup(162,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×3- 1+2
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — C6×3- 1+2
 Lower central C1 — C3 — C6×3- 1+2
 Upper central C1 — C3×C6 — C6×3- 1+2

Generators and relations for C6×3- 1+2
G = < a,b,c | a6=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Subgroups: 100 in 76 conjugacy classes, 64 normal (10 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C3×C18, C2×3- 1+2, C32×C6, C3×3- 1+2, C6×3- 1+2
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C6×3- 1+2

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

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

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

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

C6×3- 1+2 is a maximal subgroup of   C33.Dic3

66 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 6I ··· 6N 9A ··· 9R 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 3- 1+2 C2×3- 1+2 kernel C6×3- 1+2 C3×3- 1+2 C3×C18 C2×3- 1+2 C32×C6 C3×C9 3- 1+2 C33 C6 C3 # reps 1 1 6 18 2 6 18 2 6 6

Matrix representation of C6×3- 1+2 in GL4(𝔽19) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 7 0 0 0 0 1 0 18 0 0 0 11 0 13 8 18
,
 1 0 0 0 0 11 12 7 0 0 1 0 0 0 0 7
G:=sub<GL(4,GF(19))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[7,0,0,0,0,1,0,13,0,0,0,8,0,18,11,18],[1,0,0,0,0,11,0,0,0,12,1,0,0,7,0,7] >;

C6×3- 1+2 in GAP, Magma, Sage, TeX

C_6\times 3_-^{1+2}
% in TeX

G:=Group("C6xES-(3,1)");
// GroupNames label

G:=SmallGroup(162,49);
// by ID

G=gap.SmallGroup(162,49);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,457]);
// Polycyclic

G:=Group<a,b,c|a^6=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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