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## G = He3⋊C4order 108 = 22·33

### The semidirect product of He3 and C4 acting faithfully

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C4
 Chief series C1 — C3 — He3 — He3⋊C2 — He3⋊C4
 Lower central He3 — He3⋊C4
 Upper central C1 — C3

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 >

Character table of He3⋊C4

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 12A 12B 12C 12D size 1 9 1 1 12 12 9 9 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 1 1 1 -i i -1 -1 i i -i -i linear of order 4 ρ4 1 -1 1 1 1 1 i -i -1 -1 -i -i i i linear of order 4 ρ5 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 1 1 ζ6 ζ65 ζ32 ζ3 ζ3 ζ32 complex faithful ρ6 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 1 1 ζ65 ζ6 ζ3 ζ32 ζ32 ζ3 complex faithful ρ7 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 -1 -1 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 complex faithful ρ8 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 -1 -1 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 complex faithful ρ9 3 1 -3+3√-3/2 -3-3√-3/2 0 0 i -i ζ3 ζ32 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ3 complex faithful ρ10 3 1 -3-3√-3/2 -3+3√-3/2 0 0 -i i ζ32 ζ3 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ32 complex faithful ρ11 3 1 -3-3√-3/2 -3+3√-3/2 0 0 i -i ζ32 ζ3 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ32 complex faithful ρ12 3 1 -3+3√-3/2 -3-3√-3/2 0 0 -i i ζ3 ζ32 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ3 complex faithful ρ13 4 0 4 4 1 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ14 4 0 4 4 -2 1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4

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

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

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

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

 0 5 1 0 2 3 6 6 5
,
 2 0 0 0 2 0 0 0 2
,
 0 0 1 0 2 3 3 0 5
,
 0 1 3 4 3 5 0 5 3
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

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