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G = C3×He3⋊C2  order 162 = 2·34

Direct product of C3 and He3⋊C2

direct product, non-abelian, supersoluble, monomial

Aliases: C3×He3⋊C2, He3⋊5C6, C33⋊5S3, (C3×He3)⋊4C2, C32⋊2(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 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q3R3S3T6A6B6C6D6E6F6G6H
 size 191111111166666666666699999999
ρ1111111111111111111111111111111    trivial
ρ21-111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ3ζ321ζ321ζ3ζ3ζ3ζ3111ζ32ζ32ζ32ζ32111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ41-1ζ32ζ3ζ3ζ3ζ321ζ321ζ3ζ3ζ3ζ3111ζ32ζ32ζ32ζ321-1-1ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ511ζ3ζ32ζ32ζ32ζ31ζ31ζ32ζ32ζ32ζ32111ζ3ζ3ζ3ζ3111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ61-1ζ3ζ32ζ32ζ32ζ31ζ31ζ32ζ32ζ32ζ32111ζ3ζ3ζ3ζ31-1-1ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ72022222222-12-1-1-1-12-12-1-1-100000000    orthogonal lifted from S3
ρ82022222222-1-12-1-1-1-1-1-12-1200000000    orthogonal lifted from S3
ρ920222222222-1-1-1-12-12-1-1-1-100000000    orthogonal lifted from S3
ρ102022222222-1-1-122-1-1-1-1-12-100000000    orthogonal lifted from S3
ρ1120-1+√-3-1-√-3-1-√-3-1-√-3-1+√-32-1+√-32ζ6ζ6ζ6-1-√-32-1-1ζ65ζ65ζ65-1+√-3-100000000    complex lifted from C3×S3
ρ1220-1+√-3-1-√-3-1-√-3-1-√-3-1+√-32-1+√-32-1-√-3ζ6ζ6ζ6-12-1-1+√-3ζ65ζ65ζ65-100000000    complex lifted from C3×S3
ρ1320-1-√-3-1+√-3-1+√-3-1+√-3-1-√-32-1-√-32ζ65ζ65ζ65-1+√-32-1-1ζ6ζ6ζ6-1-√-3-100000000    complex lifted from C3×S3
ρ1420-1-√-3-1+√-3-1+√-3-1+√-3-1-√-32-1-√-32ζ65-1+√-3ζ65ζ65-1-12ζ6-1-√-3ζ6ζ6-100000000    complex lifted from C3×S3
ρ1520-1+√-3-1-√-3-1-√-3-1-√-3-1+√-32-1+√-32ζ6-1-√-3ζ6ζ6-1-12ζ65-1+√-3ζ65ζ65-100000000    complex lifted from C3×S3
ρ1620-1-√-3-1+√-3-1+√-3-1+√-3-1-√-32-1-√-32-1+√-3ζ65ζ65ζ65-12-1-1-√-3ζ6ζ6ζ6-100000000    complex lifted from C3×S3
ρ1720-1-√-3-1+√-3-1+√-3-1+√-3-1-√-32-1-√-32ζ65ζ65-1+√-3ζ65-1-1-1ζ6ζ6-1-√-3ζ6200000000    complex lifted from C3×S3
ρ1820-1+√-3-1-√-3-1-√-3-1-√-3-1+√-32-1+√-32ζ6ζ6-1-√-3ζ6-1-1-1ζ65ζ65-1+√-3ζ65200000000    complex lifted from C3×S3
ρ19313-3-3√-3/2-3+3√-3/23-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/2000000000000ζ32ζ3ζ32ζ31ζ31ζ32    complex lifted from He3⋊C2
ρ203-1-3-3√-3/23-3-3√-3/2-3+3√-3/23-3-3√-3/2-3+3√-3/2-3+3√-3/2000000000000ζ6ζ65-1ζ6ζ65-1ζ6ζ65    complex lifted from He3⋊C2
ρ2131-3-3√-3/23-3-3√-3/2-3+3√-3/23-3-3√-3/2-3+3√-3/2-3+3√-3/2000000000000ζ32ζ31ζ32ζ31ζ32ζ3    complex lifted from He3⋊C2
ρ223-1-3+3√-3/23-3+3√-3/2-3-3√-3/23-3+3√-3/2-3-3√-3/2-3-3√-3/2000000000000ζ65ζ6-1ζ65ζ6-1ζ65ζ6    complex lifted from He3⋊C2
ρ233-1-3+3√-3/2-3+3√-3/23-3-3√-3/2-3-3√-3/2-3-3√-3/23-3+3√-3/2000000000000ζ6ζ65ζ65-1ζ6ζ6ζ65-1    complex lifted from He3⋊C2
ρ2431-3-3√-3/2-3-3√-3/23-3+3√-3/2-3+3√-3/2-3+3√-3/23-3-3√-3/2000000000000ζ3ζ32ζ321ζ3ζ3ζ321    complex lifted from He3⋊C2
ρ25313-3+3√-3/2-3-3√-3/23-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/2000000000000ζ3ζ32ζ3ζ321ζ321ζ3    complex lifted from He3⋊C2
ρ2631-3+3√-3/2-3+3√-3/23-3-3√-3/2-3-3√-3/2-3-3√-3/23-3+3√-3/2000000000000ζ32ζ3ζ31ζ32ζ32ζ31    complex lifted from He3⋊C2
ρ273-13-3-3√-3/2-3+3√-3/23-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/2000000000000ζ6ζ65ζ6ζ65-1ζ65-1ζ6    complex lifted from He3⋊C2
ρ283-13-3+3√-3/2-3-3√-3/23-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/2000000000000ζ65ζ6ζ65ζ6-1ζ6-1ζ65    complex lifted from He3⋊C2
ρ2931-3+3√-3/23-3+3√-3/2-3-3√-3/23-3+3√-3/2-3-3√-3/2-3-3√-3/2000000000000ζ3ζ321ζ3ζ321ζ3ζ32    complex lifted from He3⋊C2
ρ303-1-3-3√-3/2-3-3√-3/23-3+3√-3/2-3+3√-3/2-3+3√-3/23-3-3√-3/2000000000000ζ65ζ6ζ6-1ζ65ζ65ζ6-1    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
 He3⋊4Dic3  He3⋊6D6  He3⋊C18  C3≀C3⋊C6  He3.C3⋊C6  He3.(C3×C6)  C34⋊6S3  3+ 1+4⋊2C2  3- 1+4⋊2C2
C3×He3⋊C2 is a maximal quotient of
 C34⋊3S3  C34.7S3  (C32×C9)⋊S3  C33⋊(C3×S3)  He3.C3⋊2C6  He3⋊(C3×S3)  C3.He3⋊C6

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

20000
02000
00100
00010
00001
,
10000
01000
00010
00001
00100
,
10000
01000
00400
00040
00004
,
66000
10000
00002
00400
00010
,
10000
66000
00100
00001
00010

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

Character table of C3×He3⋊C2 in TeX

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