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## G = C2×He3⋊5S3order 324 = 22·34

### Direct product of C2 and He3⋊5S3

Aliases: C2×He35S3, He310D6, C3312D6, C6⋊(He3⋊C2), (C2×He3)⋊5S3, (C6×He3)⋊5C2, (C32×C6)⋊6S3, (C3×He3)⋊10C22, C6.5(C33⋊C2), (C3×C6)⋊2(C3⋊S3), C323(C2×C3⋊S3), C32(C2×He3⋊C2), C3.2(C2×C33⋊C2), SmallGroup(324,150)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — C2×He3⋊5S3
 Chief series C1 — C3 — C32 — He3 — C3×He3 — He3⋊5S3 — C2×He3⋊5S3
 Lower central C3×He3 — C2×He3⋊5S3
 Upper central C1 — C6

Generators and relations for C2×He35S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, dbd-1=bc-1, fbf=b-1, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1228 in 240 conjugacy classes, 63 normal (11 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, He3, C33, S3×C6, C2×C3⋊S3, He3⋊C2, C2×He3, C3×C3⋊S3, C32×C6, C3×He3, C2×He3⋊C2, C6×C3⋊S3, He35S3, C6×He3, C2×He35S3
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, He3⋊C2, C33⋊C2, C2×He3⋊C2, C2×C33⋊C2, He35S3, C2×He35S3

Smallest permutation representation of C2×He35S3
On 36 points
Generators in S36
(1 16)(2 17)(3 18)(4 33)(5 31)(6 32)(7 30)(8 28)(9 29)(10 22)(11 23)(12 24)(13 19)(14 20)(15 21)(25 34)(26 35)(27 36)
(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)
(1 14 10)(2 15 11)(3 13 12)(4 9 36)(5 7 34)(6 8 35)(16 20 22)(17 21 23)(18 19 24)(25 31 30)(26 32 28)(27 33 29)
(1 10 14)(3 13 12)(4 36 9)(6 8 35)(16 22 20)(18 19 24)(26 32 28)(27 29 33)
(2 11 15)(3 13 12)(4 36 9)(5 7 34)(17 23 21)(18 19 24)(25 31 30)(27 29 33)
(1 6)(2 5)(3 4)(7 15)(8 14)(9 13)(10 35)(11 34)(12 36)(16 32)(17 31)(18 33)(19 29)(20 28)(21 30)(22 26)(23 25)(24 27)

G:=sub<Sym(36)| (1,16)(2,17)(3,18)(4,33)(5,31)(6,32)(7,30)(8,28)(9,29)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(25,34)(26,35)(27,36), (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), (1,14,10)(2,15,11)(3,13,12)(4,9,36)(5,7,34)(6,8,35)(16,20,22)(17,21,23)(18,19,24)(25,31,30)(26,32,28)(27,33,29), (1,10,14)(3,13,12)(4,36,9)(6,8,35)(16,22,20)(18,19,24)(26,32,28)(27,29,33), (2,11,15)(3,13,12)(4,36,9)(5,7,34)(17,23,21)(18,19,24)(25,31,30)(27,29,33), (1,6)(2,5)(3,4)(7,15)(8,14)(9,13)(10,35)(11,34)(12,36)(16,32)(17,31)(18,33)(19,29)(20,28)(21,30)(22,26)(23,25)(24,27)>;

G:=Group( (1,16)(2,17)(3,18)(4,33)(5,31)(6,32)(7,30)(8,28)(9,29)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(25,34)(26,35)(27,36), (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), (1,14,10)(2,15,11)(3,13,12)(4,9,36)(5,7,34)(6,8,35)(16,20,22)(17,21,23)(18,19,24)(25,31,30)(26,32,28)(27,33,29), (1,10,14)(3,13,12)(4,36,9)(6,8,35)(16,22,20)(18,19,24)(26,32,28)(27,29,33), (2,11,15)(3,13,12)(4,36,9)(5,7,34)(17,23,21)(18,19,24)(25,31,30)(27,29,33), (1,6)(2,5)(3,4)(7,15)(8,14)(9,13)(10,35)(11,34)(12,36)(16,32)(17,31)(18,33)(19,29)(20,28)(21,30)(22,26)(23,25)(24,27) );

G=PermutationGroup([[(1,16),(2,17),(3,18),(4,33),(5,31),(6,32),(7,30),(8,28),(9,29),(10,22),(11,23),(12,24),(13,19),(14,20),(15,21),(25,34),(26,35),(27,36)], [(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)], [(1,14,10),(2,15,11),(3,13,12),(4,9,36),(5,7,34),(6,8,35),(16,20,22),(17,21,23),(18,19,24),(25,31,30),(26,32,28),(27,33,29)], [(1,10,14),(3,13,12),(4,36,9),(6,8,35),(16,22,20),(18,19,24),(26,32,28),(27,29,33)], [(2,11,15),(3,13,12),(4,36,9),(5,7,34),(17,23,21),(18,19,24),(25,31,30),(27,29,33)], [(1,6),(2,5),(3,4),(7,15),(8,14),(9,13),(10,35),(11,34),(12,36),(16,32),(17,31),(18,33),(19,29),(20,28),(21,30),(22,26),(23,25),(24,27)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F ··· 3Q 6A 6B 6C 6D 6E 6F ··· 6Q 6R 6S 6T 6U order 1 2 2 2 3 3 3 3 3 3 ··· 3 6 6 6 6 6 6 ··· 6 6 6 6 6 size 1 1 27 27 1 1 2 2 2 6 ··· 6 1 1 2 2 2 6 ··· 6 27 27 27 27

42 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + image C1 C2 C2 S3 S3 D6 D6 He3⋊C2 C2×He3⋊C2 He3⋊5S3 C2×He3⋊5S3 kernel C2×He3⋊5S3 He3⋊5S3 C6×He3 C2×He3 C32×C6 He3 C33 C6 C3 C2 C1 # reps 1 2 1 9 4 9 4 4 4 2 2

Matrix representation of C2×He35S3 in GL5(𝔽7)

 6 0 0 0 0 0 6 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 1 0 0 0 0 2 2 2 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 6 3 4 0 0 0 1 0 0 0 5 0 0
,
 0 6 0 0 0 1 6 0 0 0 0 0 0 0 3 0 0 0 1 0 0 0 2 1 6
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 2 1 6

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

C2×He35S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_5S_3
% in TeX

G:=Group("C2xHe3:5S3");
// GroupNames label

G:=SmallGroup(324,150);
// by ID

G=gap.SmallGroup(324,150);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,579,2164,382]);
// Polycyclic

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

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