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G = He3.C6order 162 = 2·34

1st non-split extension by He3 of C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.1C6, (C3×C9)⋊2S3, He3⋊C2.C3, He3.C33C2, C32.2(C3×S3), C3.7(C32⋊C6), SmallGroup(162,12)

Series: Derived Chief Lower central Upper central

C1C3He3 — He3.C6
C1C3C32He3He3.C3 — He3.C6
He3 — He3.C6
C1C3

Generators and relations for He3.C6
 G = < a,b,c,d | a3=b3=c3=1, d6=b-1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >

9C2
3C3
9C3
3S3
9S3
9C6
3C32
3C9
6C9
3C3×S3
9C18
9C3×S3
23- 1+2
3S3×C9

Character table of He3.C6

 class 123A3B3C3D6A6B9A9B9C9D9E9F9G9H18A18B18C18D18E18F
 size 1911618993333331818999999
ρ11111111111111111111111    trivial
ρ21-11111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ311111111ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ411111111ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ51-11111-1-1ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ61-11111-1-1ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ720222-100222222-1-1000000    orthogonal lifted from S3
ρ820222-100-1--3-1--3-1+-3-1+-3-1+-3-1--3ζ6ζ65000000    complex lifted from C3×S3
ρ920222-100-1+-3-1+-3-1--3-1--3-1--3-1+-3ζ65ζ6000000    complex lifted from C3×S3
ρ103-1-3+3-3/2-3-3-3/200ζ65ζ6ζ94+2ζ9ζ97+2ζ9498959592ζ98+2ζ929790094929899795    complex faithful
ρ1131-3-3-3/2-3+3-3/200ζ32ζ398959592ζ94+2ζ9ζ97+2ζ94979ζ98+2ζ9200ζ95ζ97ζ9ζ98ζ92ζ94    complex faithful
ρ123-1-3+3-3/2-3-3-3/200ζ65ζ6ζ97+2ζ949799592ζ98+2ζ929895ζ94+2ζ90097989594992    complex faithful
ρ133-1-3-3-3/2-3+3-3/200ζ6ζ65ζ98+2ζ929895979ζ94+2ζ9ζ97+2ζ9495920098949792959    complex faithful
ρ143-1-3+3-3/2-3-3-3/200ζ65ζ6979ζ94+2ζ9ζ98+2ζ9298959592ζ97+2ζ940099592979498    complex faithful
ρ153-1-3-3-3/2-3+3-3/200ζ6ζ6598959592ζ94+2ζ9ζ97+2ζ94979ζ98+2ζ920095979989294    complex faithful
ρ1631-3+3-3/2-3-3-3/200ζ3ζ32979ζ94+2ζ9ζ98+2ζ9298959592ζ97+2ζ9400ζ9ζ95ζ92ζ97ζ94ζ98    complex faithful
ρ1731-3+3-3/2-3-3-3/200ζ3ζ32ζ94+2ζ9ζ97+2ζ9498959592ζ98+2ζ9297900ζ94ζ92ζ98ζ9ζ97ζ95    complex faithful
ρ1831-3-3-3/2-3+3-3/200ζ32ζ3ζ98+2ζ929895979ζ94+2ζ9ζ97+2ζ94959200ζ98ζ94ζ97ζ92ζ95ζ9    complex faithful
ρ1931-3-3-3/2-3+3-3/200ζ32ζ39592ζ98+2ζ92ζ97+2ζ94979ζ94+2ζ9989500ζ92ζ9ζ94ζ95ζ98ζ97    complex faithful
ρ2031-3+3-3/2-3-3-3/200ζ3ζ32ζ97+2ζ949799592ζ98+2ζ929895ζ94+2ζ900ζ97ζ98ζ95ζ94ζ9ζ92    complex faithful
ρ213-1-3-3-3/2-3+3-3/200ζ6ζ659592ζ98+2ζ92ζ97+2ζ94979ζ94+2ζ998950092994959897    complex faithful
ρ226066-300000000000000000    orthogonal lifted from C32⋊C6

Permutation representations of He3.C6
On 27 points - transitive group 27T45
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(11 23 17)(13 25 19)(15 27 21)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 25 16)(2 11 20)(3 15 24)(4 19 10)(5 23 14)(6 27 18)(7 13 22)(8 17 26)(9 21 12)
(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)

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

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

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

G:=TransitiveGroup(27,45);

On 27 points - transitive group 27T69
Generators in S27
(1 25 22)(2 17 14)(3 27 24)(4 19 16)(5 11 26)(6 21 18)(7 13 10)(8 23 20)(9 15 12)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 25 16)(2 14 23)(4 19 10)(5 26 17)(7 13 22)(8 20 11)(12 18 24)(15 27 21)
(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)

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

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

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

G:=TransitiveGroup(27,69);

He3.C6 is a maximal subgroup of   He3.D6  C3≀S33C3  He3.C3⋊C6  C3≀C3.C6  He3.C3⋊S3
He3.C6 is a maximal quotient of   He3.C12  C32⋊C9⋊S3  C33.(C3×S3)  C322D9.C3  (C3×C9)⋊D9  He3⋊C18  He3.C3⋊S3

Matrix representation of He3.C6 in GL3(𝔽19) generated by

010
001
100
,
700
070
007
,
100
0110
007
,
18818
181212
8812
G:=sub<GL(3,GF(19))| [0,0,1,1,0,0,0,1,0],[7,0,0,0,7,0,0,0,7],[1,0,0,0,11,0,0,0,7],[18,18,8,8,12,8,18,12,12] >;

He3.C6 in GAP, Magma, Sage, TeX

{\rm He}_3.C_6
% in TeX

G:=Group("He3.C6");
// GroupNames label

G:=SmallGroup(162,12);
// by ID

G=gap.SmallGroup(162,12);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,182,187,2883,433]);
// Polycyclic

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

Export

Subgroup lattice of He3.C6 in TeX
Character table of He3.C6 in TeX

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