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## G = C6×He3order 162 = 2·34

### Direct product of C6 and He3

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

Aliases: C6×He3, C336C6, C6.1C33, (C3×C6)⋊C32, (C32×C6)⋊2C3, C322(C3×C6), C3.1(C32×C6), SmallGroup(162,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×He3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C6×He3
 Lower central C1 — C3 — C6×He3
 Upper central C1 — C3×C6 — C6×He3

Generators and relations for C6×He3
G = < a,b,c,d | a6=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 208 in 112 conjugacy classes, 64 normal (8 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C32, C32, C32, C3×C6, C3×C6, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3

Smallest permutation representation of C6×He3
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 19 53)(2 20 54)(3 21 49)(4 22 50)(5 23 51)(6 24 52)(7 42 15)(8 37 16)(9 38 17)(10 39 18)(11 40 13)(12 41 14)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(43 45 47)(44 46 48)
(1 23 49)(2 24 50)(3 19 51)(4 20 52)(5 21 53)(6 22 54)(7 17 40)(8 18 41)(9 13 42)(10 14 37)(11 15 38)(12 16 39)(25 34 43)(26 35 44)(27 36 45)(28 31 46)(29 32 47)(30 33 48)
(1 27 38)(2 28 39)(3 29 40)(4 30 41)(5 25 42)(6 26 37)(7 19 32)(8 20 33)(9 21 34)(10 22 35)(11 23 36)(12 24 31)(13 53 43)(14 54 44)(15 49 45)(16 50 46)(17 51 47)(18 52 48)

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

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

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

C6×He3 is a maximal subgroup of   C334C12  He36Dic3

66 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AF 6A ··· 6H 6I ··· 6AF order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

66 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 kernel C6×He3 C3×He3 C2×He3 C32×C6 He3 C33 C6 C3 # reps 1 1 18 8 18 8 6 6

Matrix representation of C6×He3 in GL4(𝔽7) generated by

 2 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 2 0 0 0 0 1 0 3 0 0 2 1 0 0 0 4
,
 1 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 0 0 0 0 3 1 0 0 5 0 0 0 3 0 4
G:=sub<GL(4,GF(7))| [2,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[2,0,0,0,0,1,0,0,0,0,2,0,0,3,1,4],[1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,3,5,3,0,1,0,0,0,0,0,4] >;

C6×He3 in GAP, Magma, Sage, TeX

C_6\times {\rm He}_3
% in TeX

G:=Group("C6xHe3");
// GroupNames label

G:=SmallGroup(162,48);
// by ID

G=gap.SmallGroup(162,48);
# by ID

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

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

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