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## G = C2×C3.He3order 162 = 2·34

### Direct product of C2 and C3.He3

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

Aliases: C2×C3.He3, C6.5He3, 3- 1+2.C6, (C3×C9).3C6, (C3×C18).2C3, C3.5(C2×He3), (C3×C6).4C32, C32.4(C3×C6), (C2×3- 1+2).C3, SmallGroup(162,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3.He3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — C2×C3.He3
 Lower central C1 — C3 — C32 — C2×C3.He3
 Upper central C1 — C6 — C3×C6 — C2×C3.He3

Generators and relations for C2×C3.He3
G = < a,b,c,d,e | a2=b3=d3=1, c3=b-1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=bcd-1, ede-1=b-1d >

Smallest permutation representation of C2×C3.He3
On 54 points
Generators in S54
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(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)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(1 26 11 7 23 17 4 20 14)(2 21 12 8 27 18 5 24 15)(3 25 13 9 22 10 6 19 16)(28 46 43 34 52 40 31 49 37)(29 50 44 35 47 41 32 53 38)(30 54 45 36 51 42 33 48 39)

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

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

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

C2×C3.He3 is a maximal subgroup of   3- 1+2.Dic3

34 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 18A ··· 18F 18G ··· 18L order 1 2 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 1 1 3 3 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

34 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 C3.He3 C2×C3.He3 kernel C2×C3.He3 C3.He3 C3×C18 C2×3- 1+2 C3×C9 3- 1+2 C6 C3 C2 C1 # reps 1 1 2 6 2 6 2 2 6 6

Matrix representation of C2×C3.He3 in GL4(𝔽19) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 6 0 0 0 0 6 0 0 6 0 9
,
 1 0 0 0 0 1 0 0 0 1 11 0 0 12 0 7
,
 1 0 0 0 0 1 10 0 0 0 18 1 0 1 18 0
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,6,0,6,0,0,6,0,0,0,0,9],[1,0,0,0,0,1,1,12,0,0,11,0,0,0,0,7],[1,0,0,0,0,1,0,1,0,10,18,18,0,0,1,0] >;

C2×C3.He3 in GAP, Magma, Sage, TeX

C_2\times C_3.{\rm He}_3
% in TeX

G:=Group("C2xC3.He3");
// GroupNames label

G:=SmallGroup(162,31);
// by ID

G=gap.SmallGroup(162,31);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,187,282,728]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=d^3=1,c^3=b^-1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b*c*d^-1,e*d*e^-1=b^-1*d>;
// generators/relations

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