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G = He3⋊C3order 81 = 34

2nd semidirect product of He3 and C3 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: He32C3, C3.4He3, C32.3C32, (C3×C9)⋊3C3, 3-Sylow(PSL(3,19)), SmallGroup(81,9)

Series: Derived Chief Lower central Upper central Jennings

C1C32 — He3⋊C3
C1C3C32C3×C9 — He3⋊C3
C1C3C32 — He3⋊C3
C1C3C32 — He3⋊C3
C1C3C32 — He3⋊C3

Generators and relations for He3⋊C3
 G = < a,b,c,d | a3=b3=c3=d3=1, ab=ba, cac-1=dad-1=ab-1, bc=cb, bd=db, dcd-1=abc >

3C3
9C3
9C3
9C3
3C32
3C32
3C9
3C32

Character table of He3⋊C3

 class 13A3B3C3D3E3F3G3H3I3J9A9B9C9D9E9F
 size 11133999999333333
ρ111111111111111111    trivial
ρ211111ζ321ζ3ζ321ζ3ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ311111ζ32ζ3ζ3ζ3ζ32ζ32111111    linear of order 3
ρ4111111ζ321ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ511111ζ31ζ32ζ31ζ32ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ611111ζ3ζ3ζ321ζ321ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ711111ζ3ζ32ζ32ζ32ζ3ζ3111111    linear of order 3
ρ811111ζ32ζ32ζ31ζ31ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ9111111ζ31ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ10333-3-3-3/2-3+3-3/2000000000000    complex lifted from He3
ρ11333-3+3-3/2-3-3-3/2000000000000    complex lifted from He3
ρ123-3-3-3/2-3+3-3/200000000ζ98+2ζ95ζ97+2ζ99892ζ95+2ζ929499794    complex faithful
ρ133-3+3-3/2-3-3-3/200000000ζ97+2ζ9ζ95+2ζ9297949499892ζ98+2ζ95    complex faithful
ρ143-3-3-3/2-3+3-3/200000000ζ95+2ζ92949ζ98+2ζ9598929794ζ97+2ζ9    complex faithful
ρ153-3-3-3/2-3+3-3/20000000098929794ζ95+2ζ92ζ98+2ζ95ζ97+2ζ9949    complex faithful
ρ163-3+3-3/2-3-3-3/2000000009499892ζ97+2ζ99794ζ98+2ζ95ζ95+2ζ92    complex faithful
ρ173-3+3-3/2-3-3-3/2000000009794ζ98+2ζ95949ζ97+2ζ9ζ95+2ζ929892    complex faithful

Permutation representations of He3⋊C3
On 27 points - transitive group 27T23
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 14 10)(2 15 11)(3 13 12)(4 27 8)(5 25 9)(6 26 7)(16 23 19)(17 24 20)(18 22 21)
(2 15 11)(3 12 13)(4 9 7)(5 6 27)(8 25 26)(16 18 20)(17 23 22)(19 21 24)
(1 5 20)(2 26 22)(3 8 16)(4 23 13)(6 18 11)(7 21 15)(9 24 10)(12 27 19)(14 25 17)

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

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

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

G:=TransitiveGroup(27,23);

On 27 points - transitive group 27T24
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 27 26)
(1 25 14)(2 26 13)(3 27 15)(4 21 17)(5 19 16)(6 20 18)(7 24 11)(8 22 10)(9 23 12)
(1 24 17)(2 22 16)(3 23 18)(4 27 11)(5 25 10)(6 26 12)(7 20 15)(8 21 14)(9 19 13)

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

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

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

G:=TransitiveGroup(27,24);

He3⋊C3 is a maximal subgroup of
He3.2C6  He3.2S3  He3⋊S3  C92⋊C3  C922C3  C32.He3  C32.6He3  C9.He3  He3⋊C32  C9.2He3  C62.14C32  He3⋊A4
He3⋊C3 is a maximal quotient of
C32.24He3  C32.27He3  C32.29He3  C32.20He3  He3⋊C9  C92⋊C3  C922C3  C92.C3  C32.He3  C32.5He3  C32.6He3  C62.14C32  He3⋊A4

Matrix representation of He3⋊C3 in GL3(𝔽19) generated by

160
0181
0180
,
700
070
007
,
649
10139
6130
,
704
18012
71112
G:=sub<GL(3,GF(19))| [1,0,0,6,18,18,0,1,0],[7,0,0,0,7,0,0,0,7],[6,10,6,4,13,13,9,9,0],[7,18,7,0,0,11,4,12,12] >;

He3⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("He3:C3");
// GroupNames label

G:=SmallGroup(81,9);
// by ID

G=gap.SmallGroup(81,9);
# by ID

G:=PCGroup([4,-3,3,-3,-3,97,149,434]);
// Polycyclic

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

Export

Subgroup lattice of He3⋊C3 in TeX
Character table of He3⋊C3 in TeX

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