Copied to
clipboard

## G = C3×He3⋊C2order 162 = 2·34

### Direct product of C3 and He3⋊C2

Aliases: C3×He3⋊C2, He35C6, C335S3, (C3×He3)⋊4C2, C322(C3×S3), C32.11(C3⋊S3), C3.6(C3×C3⋊S3), SmallGroup(162,41)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×He3⋊C2
G = < a,b,c,d,e | a3=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: 232 in 76 conjugacy classes, 18 normal (7 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3×C6, He3, He3, C33, He3⋊C2, S3×C32, C3×He3, C3×He3⋊C2
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, C3×He3⋊C2

Character table of C3×He3⋊C2

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T 6A 6B 6C 6D 6E 6F 6G 6H size 1 9 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ4 1 -1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ32 1 -1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ5 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ6 1 -1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ3 1 -1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ7 2 0 2 2 2 2 2 2 2 2 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 2 2 2 2 2 2 -1 -1 2 -1 -1 -1 -1 -1 -1 2 -1 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ9 2 0 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ10 2 0 2 2 2 2 2 2 2 2 -1 -1 -1 2 2 -1 -1 -1 -1 -1 2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ11 2 0 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 2 -1+√-3 2 ζ6 ζ6 ζ6 -1-√-3 2 -1 -1 ζ65 ζ65 ζ65 -1+√-3 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ12 2 0 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 2 -1+√-3 2 -1-√-3 ζ6 ζ6 ζ6 -1 2 -1 -1+√-3 ζ65 ζ65 ζ65 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ13 2 0 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 2 -1-√-3 2 ζ65 ζ65 ζ65 -1+√-3 2 -1 -1 ζ6 ζ6 ζ6 -1-√-3 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ14 2 0 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 2 -1-√-3 2 ζ65 -1+√-3 ζ65 ζ65 -1 -1 2 ζ6 -1-√-3 ζ6 ζ6 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ15 2 0 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 2 -1+√-3 2 ζ6 -1-√-3 ζ6 ζ6 -1 -1 2 ζ65 -1+√-3 ζ65 ζ65 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ16 2 0 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 2 -1-√-3 2 -1+√-3 ζ65 ζ65 ζ65 -1 2 -1 -1-√-3 ζ6 ζ6 ζ6 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ17 2 0 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 2 -1-√-3 2 ζ65 ζ65 -1+√-3 ζ65 -1 -1 -1 ζ6 ζ6 -1-√-3 ζ6 2 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ18 2 0 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 2 -1+√-3 2 ζ6 ζ6 -1-√-3 ζ6 -1 -1 -1 ζ65 ζ65 -1+√-3 ζ65 2 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ19 3 1 3 -3-3√-3/2 -3+3√-3/2 3 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 complex lifted from He3⋊C2 ρ20 3 -1 -3-3√-3/2 3 -3-3√-3/2 -3+3√-3/2 3 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ6 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ65 complex lifted from He3⋊C2 ρ21 3 1 -3-3√-3/2 3 -3-3√-3/2 -3+3√-3/2 3 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 complex lifted from He3⋊C2 ρ22 3 -1 -3+3√-3/2 3 -3+3√-3/2 -3-3√-3/2 3 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ65 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ6 complex lifted from He3⋊C2 ρ23 3 -1 -3+3√-3/2 -3+3√-3/2 3 -3-3√-3/2 -3-3√-3/2 -3-3√-3/2 3 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ6 ζ65 ζ65 -1 ζ6 ζ6 ζ65 -1 complex lifted from He3⋊C2 ρ24 3 1 -3-3√-3/2 -3-3√-3/2 3 -3+3√-3/2 -3+3√-3/2 -3+3√-3/2 3 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 complex lifted from He3⋊C2 ρ25 3 1 3 -3+3√-3/2 -3-3√-3/2 3 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 complex lifted from He3⋊C2 ρ26 3 1 -3+3√-3/2 -3+3√-3/2 3 -3-3√-3/2 -3-3√-3/2 -3-3√-3/2 3 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 complex lifted from He3⋊C2 ρ27 3 -1 3 -3-3√-3/2 -3+3√-3/2 3 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 complex lifted from He3⋊C2 ρ28 3 -1 3 -3+3√-3/2 -3-3√-3/2 3 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 complex lifted from He3⋊C2 ρ29 3 1 -3+3√-3/2 3 -3+3√-3/2 -3-3√-3/2 3 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 complex lifted from He3⋊C2 ρ30 3 -1 -3-3√-3/2 -3-3√-3/2 3 -3+3√-3/2 -3+3√-3/2 -3+3√-3/2 3 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 ζ65 ζ6 ζ6 -1 ζ65 ζ65 ζ6 -1 complex lifted from He3⋊C2

Permutation representations of C3×He3⋊C2
On 27 points - transitive group 27T46
Generators in S27
(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)
(1 20 10)(2 21 11)(3 19 12)(4 25 15)(5 26 13)(6 27 14)(7 24 17)(8 22 18)(9 23 16)
(1 5 22)(2 6 23)(3 4 24)(7 12 15)(8 10 13)(9 11 14)(16 21 27)(17 19 25)(18 20 26)
(7 15 12)(8 13 10)(9 14 11)(16 21 27)(17 19 25)(18 20 26)
(7 17)(8 18)(9 16)(10 20)(11 21)(12 19)(13 26)(14 27)(15 25)

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

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), (1,20,10)(2,21,11)(3,19,12)(4,25,15)(5,26,13)(6,27,14)(7,24,17)(8,22,18)(9,23,16), (1,5,22)(2,6,23)(3,4,24)(7,12,15)(8,10,13)(9,11,14)(16,21,27)(17,19,25)(18,20,26), (7,15,12)(8,13,10)(9,14,11)(16,21,27)(17,19,25)(18,20,26), (7,17)(8,18)(9,16)(10,20)(11,21)(12,19)(13,26)(14,27)(15,25) );

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)], [(1,20,10),(2,21,11),(3,19,12),(4,25,15),(5,26,13),(6,27,14),(7,24,17),(8,22,18),(9,23,16)], [(1,5,22),(2,6,23),(3,4,24),(7,12,15),(8,10,13),(9,11,14),(16,21,27),(17,19,25),(18,20,26)], [(7,15,12),(8,13,10),(9,14,11),(16,21,27),(17,19,25),(18,20,26)], [(7,17),(8,18),(9,16),(10,20),(11,21),(12,19),(13,26),(14,27),(15,25)]])

G:=TransitiveGroup(27,46);

C3×He3⋊C2 is a maximal subgroup of
He34Dic3  He36D6  He3⋊C18  C3≀C3⋊C6  He3.C3⋊C6  He3.(C3×C6)  C346S3  3+ 1+42C2  3- 1+42C2
C3×He3⋊C2 is a maximal quotient of
C343S3  C34.7S3  (C32×C9)⋊S3  C33⋊(C3×S3)  He3.C32C6  He3⋊(C3×S3)  C3.He3⋊C6

Matrix representation of C3×He3⋊C2 in GL5(𝔽7)

 2 0 0 0 0 0 2 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 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 6 6 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 4 0 0 0 0 0 1 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))| [2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[6,1,0,0,0,6,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,2,0,0],[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] >;

C3×He3⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C3xHe3:C2");
// GroupNames label

G:=SmallGroup(162,41);
// by ID

G=gap.SmallGroup(162,41);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,182,723,253]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=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

Export

׿
×
𝔽