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## G = C6×He3⋊C2order 324 = 22·34

### Direct product of C6 and He3⋊C2

Aliases: C6×He3⋊C2, C3311D6, (C6×He3)⋊4C2, He36(C2×C6), (C2×He3)⋊5C6, (C32×C6)⋊5S3, C324(S3×C6), (C3×He3)⋊9C22, (C3×C6)⋊2(C3×S3), C3.6(C6×C3⋊S3), C6.14(C3×C3⋊S3), (C3×C6).26(C3⋊S3), C32.13(C2×C3⋊S3), SmallGroup(324,145)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×He3⋊C2
G = < a,b,c,d,e | a6=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 592 in 184 conjugacy classes, 38 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, He3, He3, C33, S3×C6, C62, He3⋊C2, C2×He3, C2×He3, S3×C32, C32×C6, C3×He3, C2×He3⋊C2, S3×C3×C6, C3×He3⋊C2, C6×He3, C6×He3⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, He3⋊C2, C3×C3⋊S3, C2×He3⋊C2, C6×C3⋊S3, C3×He3⋊C2, C6×He3⋊C2

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3T 6A ··· 6H 6I ··· 6T 6U ··· 6AJ order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 9 9 1 ··· 1 6 ··· 6 1 ··· 1 6 ··· 6 9 ··· 9

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 He3⋊C2 C2×He3⋊C2 kernel C6×He3⋊C2 C3×He3⋊C2 C6×He3 C2×He3⋊C2 He3⋊C2 C2×He3 C32×C6 C33 C3×C6 C32 C6 C3 # reps 1 2 1 2 4 2 4 4 8 8 12 12

Matrix representation of C6×He3⋊C2 in GL5(𝔽7)

 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 6 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2
,
 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 6 6 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(7))| [3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,1,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[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],[1,6,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C6×He3⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C6xHe3:C2");
// GroupNames label

G:=SmallGroup(324,145);
// by ID

G=gap.SmallGroup(324,145);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,2164,382]);
// Polycyclic

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

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