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## G = C2×He3.C6order 324 = 22·34

### Direct product of C2 and He3.C6

Aliases: C2×He3.C6, (C3×C9)⋊6D6, (C3×C18)⋊2S3, He3⋊C2.C6, (C2×He3).4C6, He3.1(C2×C6), C32.2(S3×C6), He3.C34C22, C6.16(C32⋊C6), (C3×C6).5(C3×S3), (C2×He3⋊C2).C3, C3.7(C2×C32⋊C6), (C2×He3.C3)⋊3C2, SmallGroup(324,70)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×He3.C6
G = < a,b,c,d,e | a2=b3=c3=d3=1, e6=c-1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1c, cd=dc, ce=ec, ede-1=b-1d-1 >

Smallest permutation representation of C2×He3.C6
On 54 points
Generators in S54
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 20 41)(2 54 27)(3 22 43)(4 38 29)(5 24 45)(6 40 31)(7 26 47)(8 42 33)(9 28 49)(10 44 35)(11 30 51)(12 46 19)(13 32 53)(14 48 21)(15 34 37)(16 50 23)(17 36 39)(18 52 25)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)
(2 54 21)(3 43 28)(5 24 39)(6 31 46)(8 42 27)(9 49 34)(11 30 45)(12 19 52)(14 48 33)(15 37 22)(17 36 51)(18 25 40)(20 32 26)(23 29 35)(38 50 44)(41 47 53)
(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)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 6E 6F 6G 6H 9A ··· 9F 9G 9H 18A ··· 18F 18G ··· 18R 18S 18T order 1 2 2 2 3 3 3 3 6 6 6 6 6 6 6 6 9 ··· 9 9 9 18 ··· 18 18 ··· 18 18 18 size 1 1 9 9 1 1 6 18 1 1 6 9 9 9 9 18 3 ··· 3 18 18 3 ··· 3 9 ··· 9 18 18

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 He3.C6 C2×He3.C6 C32⋊C6 C2×C32⋊C6 kernel C2×He3.C6 He3.C6 C2×He3.C3 C2×He3⋊C2 He3⋊C2 C2×He3 C3×C18 C3×C9 C3×C6 C32 C2 C1 C6 C3 # reps 1 2 1 2 4 2 1 1 2 2 12 12 1 1

Matrix representation of C2×He3.C6 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 18 18 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 11
,
 18 0 0 0 0 1 1 0 0 0 0 0 8 8 12 0 0 12 18 12 0 0 8 18 18

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[18,1,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[18,1,0,0,0,0,1,0,0,0,0,0,8,12,8,0,0,8,18,18,0,0,12,12,18] >;

C2×He3.C6 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3.C_6
% in TeX

G:=Group("C2xHe3.C6");
// GroupNames label

G:=SmallGroup(324,70);
// by ID

G=gap.SmallGroup(324,70);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,579,303,8644,652]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=1,e^6=c^-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,e*b*e^-1=b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*d^-1>;
// generators/relations

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