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G = He3⋊C4order 108 = 22·33

The semidirect product of He3 and C4 acting faithfully

non-abelian, soluble

Aliases: He3⋊C4, C3.(C32⋊C4), He3⋊C2.C2, SmallGroup(108,15)

Series: Derived Chief Lower central Upper central

C1C3He3 — He3⋊C4
C1C3He3He3⋊C2 — He3⋊C4
He3 — He3⋊C4
C1C3

Generators and relations for He3⋊C4
 G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, cac-1=ab-1, dad-1=abc, bc=cb, bd=db, dcd-1=ac-1 >

9C2
6C3
6C3
9C4
6S3
6S3
9C6
2C32
2C32
9C12
6C3×S3
6C3×S3

Character table of He3⋊C4

 class 123A3B3C3D4A4B6A6B12A12B12C12D
 size 1911121299999999
ρ111111111111111    trivial
ρ2111111-1-111-1-1-1-1    linear of order 2
ρ31-11111-ii-1-1ii-i-i    linear of order 4
ρ41-11111i-i-1-1-i-iii    linear of order 4
ρ53-1-3-3-3/2-3+3-3/20011ζ6ζ65ζ32ζ3ζ3ζ32    complex faithful
ρ63-1-3+3-3/2-3-3-3/20011ζ65ζ6ζ3ζ32ζ32ζ3    complex faithful
ρ73-1-3+3-3/2-3-3-3/200-1-1ζ65ζ6ζ65ζ6ζ6ζ65    complex faithful
ρ83-1-3-3-3/2-3+3-3/200-1-1ζ6ζ65ζ6ζ65ζ65ζ6    complex faithful
ρ931-3+3-3/2-3-3-3/200i-iζ3ζ32ζ43ζ3ζ43ζ32ζ4ζ32ζ4ζ3    complex faithful
ρ1031-3-3-3/2-3+3-3/200-iiζ32ζ3ζ4ζ32ζ4ζ3ζ43ζ3ζ43ζ32    complex faithful
ρ1131-3-3-3/2-3+3-3/200i-iζ32ζ3ζ43ζ32ζ43ζ3ζ4ζ3ζ4ζ32    complex faithful
ρ1231-3+3-3/2-3-3-3/200-iiζ3ζ32ζ4ζ3ζ4ζ32ζ43ζ32ζ43ζ3    complex faithful
ρ1340441-200000000    orthogonal lifted from C32⋊C4
ρ144044-2100000000    orthogonal lifted from C32⋊C4

Permutation representations of He3⋊C4
On 18 points - transitive group 18T49
Generators in S18
(1 17 11)(2 3 5)(4 13 8)(6 10 15)(9 12 16)
(1 6 4)(2 5 3)(7 14 18)(8 11 15)(9 12 16)(10 13 17)
(1 11 13)(2 16 18)(3 12 14)(4 8 10)(5 9 7)(6 15 17)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

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

G:=Group( (1,17,11)(2,3,5)(4,13,8)(6,10,15)(9,12,16), (1,6,4)(2,5,3)(7,14,18)(8,11,15)(9,12,16)(10,13,17), (1,11,13)(2,16,18)(3,12,14)(4,8,10)(5,9,7)(6,15,17), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );

G=PermutationGroup([(1,17,11),(2,3,5),(4,13,8),(6,10,15),(9,12,16)], [(1,6,4),(2,5,3),(7,14,18),(8,11,15),(9,12,16),(10,13,17)], [(1,11,13),(2,16,18),(3,12,14),(4,8,10),(5,9,7),(6,15,17)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)])

G:=TransitiveGroup(18,49);

On 27 points - transitive group 27T32
Generators in S27
(1 22 15)(2 13 11)(3 9 20)(4 21 5)(6 26 14)(7 10 17)(8 24 27)(12 16 19)(18 23 25)
(1 3 2)(4 27 19)(5 24 16)(6 25 17)(7 26 18)(8 12 21)(9 13 22)(10 14 23)(11 15 20)
(1 27 25)(2 4 6)(3 19 17)(5 10 15)(7 13 8)(9 21 18)(11 16 23)(12 26 22)(14 20 24)
(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)| (1,22,15)(2,13,11)(3,9,20)(4,21,5)(6,26,14)(7,10,17)(8,24,27)(12,16,19)(18,23,25), (1,3,2)(4,27,19)(5,24,16)(6,25,17)(7,26,18)(8,12,21)(9,13,22)(10,14,23)(11,15,20), (1,27,25)(2,4,6)(3,19,17)(5,10,15)(7,13,8)(9,21,18)(11,16,23)(12,26,22)(14,20,24), (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( (1,22,15)(2,13,11)(3,9,20)(4,21,5)(6,26,14)(7,10,17)(8,24,27)(12,16,19)(18,23,25), (1,3,2)(4,27,19)(5,24,16)(6,25,17)(7,26,18)(8,12,21)(9,13,22)(10,14,23)(11,15,20), (1,27,25)(2,4,6)(3,19,17)(5,10,15)(7,13,8)(9,21,18)(11,16,23)(12,26,22)(14,20,24), (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([(1,22,15),(2,13,11),(3,9,20),(4,21,5),(6,26,14),(7,10,17),(8,24,27),(12,16,19),(18,23,25)], [(1,3,2),(4,27,19),(5,24,16),(6,25,17),(7,26,18),(8,12,21),(9,13,22),(10,14,23),(11,15,20)], [(1,27,25),(2,4,6),(3,19,17),(5,10,15),(7,13,8),(9,21,18),(11,16,23),(12,26,22),(14,20,24)], [(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(27,32);

He3⋊C4 is a maximal subgroup of   He3⋊C8  He3⋊D4  SU3(𝔽2)  He3.3C12  He34Dic3
He3⋊C4 is a maximal quotient of   He32C8  He34Dic3

Matrix representation of He3⋊C4 in GL3(𝔽7) generated by

051
023
665
,
200
020
002
,
001
023
305
,
013
435
053
G:=sub<GL(3,GF(7))| [0,0,6,5,2,6,1,3,5],[2,0,0,0,2,0,0,0,2],[0,0,3,0,2,0,1,3,5],[0,4,0,1,3,5,3,5,3] >;

He3⋊C4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_4
% in TeX

G:=Group("He3:C4");
// GroupNames label

G:=SmallGroup(108,15);
// by ID

G=gap.SmallGroup(108,15);
# by ID

G:=PCGroup([5,-2,-2,-3,3,-3,10,422,67,643,608,253]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^4=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations

Export

Subgroup lattice of He3⋊C4 in TeX
Character table of He3⋊C4 in TeX

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