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G = C2×He3⋊C2  order 108 = 22·33

Direct product of C2 and He3⋊C2

direct product, non-abelian, supersoluble, monomial

Aliases: C2×He3⋊C2, C32⋊3D6, He3⋊3C22, (C3×C6)⋊2S3, C6.5(C3⋊S3), (C2×He3)⋊2C2, C3.2(C2×C3⋊S3), SmallGroup(108,28)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×He3⋊C2
 G = < a,b,c,d,e | a2=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: 167 in 55 conjugacy classes, 17 normal (7 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, D6, C2×C6, C3×S3, C3×C6, He3, S3×C6, He3⋊C2, C2×He3, C2×He3⋊C2
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2

Character table of C2×He3⋊C2

 class 12A2B2C3A3B3C3D3E3F6A6B6C6D6E6F6G6H6I6J
 size 11991166661166669999
ρ111111111111111111111    trivial
ρ211-1-1111111111111-1-1-1-1    linear of order 2
ρ31-11-1111111-1-1-1-1-1-1-1-111    linear of order 2
ρ41-1-11111111-1-1-1-1-1-111-1-1    linear of order 2
ρ5220022-1-12-122-12-1-10000    orthogonal lifted from S3
ρ62-20022-1-12-1-2-21-2110000    orthogonal lifted from D6
ρ72-20022-1-1-12-2-2-21110000    orthogonal lifted from D6
ρ82-20022-12-1-1-2-2111-20000    orthogonal lifted from D6
ρ9220022-12-1-122-1-1-120000    orthogonal lifted from S3
ρ102-200222-1-1-1-2-211-210000    orthogonal lifted from D6
ρ112200222-1-1-122-1-12-10000    orthogonal lifted from S3
ρ12220022-1-1-12222-1-1-10000    orthogonal lifted from S3
ρ1333-1-1-3+3√-3/2-3-3√-3/20000-3-3√-3/2-3+3√-3/20000ζ65ζ6ζ6ζ65    complex lifted from He3⋊C2
ρ1433-1-1-3-3√-3/2-3+3√-3/20000-3+3√-3/2-3-3√-3/20000ζ6ζ65ζ65ζ6    complex lifted from He3⋊C2
ρ153-31-1-3+3√-3/2-3-3√-3/200003+3√-3/23-3√-3/20000ζ65ζ6ζ32ζ3    complex faithful
ρ163-3-11-3-3√-3/2-3+3√-3/200003-3√-3/23+3√-3/20000ζ32ζ3ζ65ζ6    complex faithful
ρ173-3-11-3+3√-3/2-3-3√-3/200003+3√-3/23-3√-3/20000ζ3ζ32ζ6ζ65    complex faithful
ρ183311-3-3√-3/2-3+3√-3/20000-3+3√-3/2-3-3√-3/20000ζ32ζ3ζ3ζ32    complex lifted from He3⋊C2
ρ193311-3+3√-3/2-3-3√-3/20000-3-3√-3/2-3+3√-3/20000ζ3ζ32ζ32ζ3    complex lifted from He3⋊C2
ρ203-31-1-3-3√-3/2-3+3√-3/200003-3√-3/23+3√-3/20000ζ6ζ65ζ3ζ32    complex faithful

Permutation representations of C2×He3⋊C2
►On 18 points - transitive group 18T52
Generators in S18
(1 16)(2 17)(3 18)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 14 10)(2 15 11)(3 13 12)(4 18 8)(5 16 9)(6 17 7)

(1 13 15)(2 10 3)(4 6 9)(5 18 17)(7 16 8)(11 14 12)

(1 16)(2 18)(3 17)(4 11)(5 10)(6 12)(7 13)(8 15)(9 14)
 
G:=sub<Sym(18)| (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,15)(8,13)(9,14), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,10)(2,15,11)(3,13,12)(4,18,8)(5,16,9)(6,17,7), (1,13,15)(2,10,3)(4,6,9)(5,18,17)(7,16,8)(11,14,12), (1,16)(2,18)(3,17)(4,11)(5,10)(6,12)(7,13)(8,15)(9,14)>;
 
G:=Group( (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,15)(8,13)(9,14), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,10)(2,15,11)(3,13,12)(4,18,8)(5,16,9)(6,17,7), (1,13,15)(2,10,3)(4,6,9)(5,18,17)(7,16,8)(11,14,12), (1,16)(2,18)(3,17)(4,11)(5,10)(6,12)(7,13)(8,15)(9,14) );
 
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,12),(5,10),(6,11),(7,15),(8,13),(9,14)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,14,10),(2,15,11),(3,13,12),(4,18,8),(5,16,9),(6,17,7)], [(1,13,15),(2,10,3),(4,6,9),(5,18,17),(7,16,8),(11,14,12)], [(1,16),(2,18),(3,17),(4,11),(5,10),(6,12),(7,13),(8,15),(9,14)]])
 
G:=TransitiveGroup(18,52);
 

C2×He3⋊C2 is a maximal subgroup of   He3⋊(C2×C4)  He3⋊3D4  He3⋊5D4  He3⋊7D4  C32⋊3GL2(𝔽3)
C2×He3⋊C2 is a maximal quotient of   He3⋊4Q8  He3⋊5D4  He3⋊7D4

Matrix representation of C2×He3⋊C2 ►in GL3(𝔽7) generated by

600
060
006
,
010
001
100
,
200
020
002
,
100
040
002
,
600
006
060
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[0,0,1,1,0,0,0,1,0],[2,0,0,0,2,0,0,0,2],[1,0,0,0,4,0,0,0,2],[6,0,0,0,0,6,0,6,0] >;
 

C2×He3⋊C2 in GAP, Magma, Sage, TeX 

C_2\times {\rm He}_3\rtimes C_2
 
% in TeX
 
G:=Group("C2xHe3:C2");
 
// GroupNames label
 
G:=SmallGroup(108,28);
 
// by ID
 
G=gap.SmallGroup(108,28);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-3,122,483,253]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=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

Character table of C2×He3⋊C2 in TeX

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