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## G = He3⋊C8order 216 = 23·33

### The semidirect product of He3 and C8 acting faithfully

Aliases: He3⋊C8, C3.F9, He3⋊C2.C4, He3⋊C4.1C2, SmallGroup(216,86)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C8
 Chief series C1 — C3 — He3 — He3⋊C2 — He3⋊C4 — He3⋊C8
 Lower central He3 — He3⋊C8
 Upper central C1

Generators and relations for He3⋊C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=c, bc=cb, dbd-1=b-1, dcd-1=ab-1c >

9C2
12C3
9C4
9C6
12S3
4C32
27C8
9C12
12C3×S3

Character table of He3⋊C8

 class 1 2 3A 3B 4A 4B 6 8A 8B 8C 8D 12A 12B size 1 9 2 24 9 9 18 27 27 27 27 18 18 ρ1 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 linear of order 2 ρ3 1 1 1 1 -1 -1 1 -i i -i i -1 -1 linear of order 4 ρ4 1 1 1 1 -1 -1 1 i -i i -i -1 -1 linear of order 4 ρ5 1 -1 1 1 -i i -1 ζ8 ζ83 ζ85 ζ87 -i i linear of order 8 ρ6 1 -1 1 1 -i i -1 ζ85 ζ87 ζ8 ζ83 -i i linear of order 8 ρ7 1 -1 1 1 i -i -1 ζ87 ζ85 ζ83 ζ8 i -i linear of order 8 ρ8 1 -1 1 1 i -i -1 ζ83 ζ8 ζ87 ζ85 i -i linear of order 8 ρ9 6 -2 -3 0 2 2 1 0 0 0 0 -1 -1 orthogonal faithful ρ10 6 -2 -3 0 -2 -2 1 0 0 0 0 1 1 symplectic faithful, Schur index 2 ρ11 6 2 -3 0 2i -2i -1 0 0 0 0 -i i complex faithful ρ12 6 2 -3 0 -2i 2i -1 0 0 0 0 i -i complex faithful ρ13 8 0 8 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from F9

Permutation representations of He3⋊C8
On 27 points - transitive group 27T77
Generators in S27
(1 24 14)(2 18 4)(3 8 20)(5 10 11)(6 25 17)(7 16 19)(9 23 22)(12 27 21)(13 15 26)
(1 2 3)(4 20 14)(5 15 21)(6 22 16)(7 17 23)(8 24 18)(9 19 25)(10 26 12)(11 13 27)
(1 23 13)(2 7 27)(3 17 11)(4 9 10)(5 24 16)(6 15 18)(8 22 21)(12 14 25)(19 26 20)
(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)| (1,24,14)(2,18,4)(3,8,20)(5,10,11)(6,25,17)(7,16,19)(9,23,22)(12,27,21)(13,15,26), (1,2,3)(4,20,14)(5,15,21)(6,22,16)(7,17,23)(8,24,18)(9,19,25)(10,26,12)(11,13,27), (1,23,13)(2,7,27)(3,17,11)(4,9,10)(5,24,16)(6,15,18)(8,22,21)(12,14,25)(19,26,20), (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( (1,24,14)(2,18,4)(3,8,20)(5,10,11)(6,25,17)(7,16,19)(9,23,22)(12,27,21)(13,15,26), (1,2,3)(4,20,14)(5,15,21)(6,22,16)(7,17,23)(8,24,18)(9,19,25)(10,26,12)(11,13,27), (1,23,13)(2,7,27)(3,17,11)(4,9,10)(5,24,16)(6,15,18)(8,22,21)(12,14,25)(19,26,20), (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([(1,24,14),(2,18,4),(3,8,20),(5,10,11),(6,25,17),(7,16,19),(9,23,22),(12,27,21),(13,15,26)], [(1,2,3),(4,20,14),(5,15,21),(6,22,16),(7,17,23),(8,24,18),(9,19,25),(10,26,12),(11,13,27)], [(1,23,13),(2,7,27),(3,17,11),(4,9,10),(5,24,16),(6,15,18),(8,22,21),(12,14,25),(19,26,20)], [(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(27,77);

He3⋊C8 is a maximal subgroup of   He3⋊SD16
He3⋊C8 is a maximal quotient of   He3⋊C16

Matrix representation of He3⋊C8 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 2 0 1 2 1 0 1 1 0 1 2 0 -1 -1 0 -1 -1 -1 -1 0 -1 -1 -1
,
 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 -1 0 -1 0 -1 -1 0 1 0 1 1 0
,
 0 1 1 0 2 1 1 0 2 0 1 2 0 0 1 0 0 0 0 0 0 1 0 0 -1 0 -1 -1 -1 -1 0 -1 -1 0 -1 -1
,
 0 0 -1 0 0 0 0 0 -1 1 0 0 -1 -1 -1 0 -1 -2 -1 -1 0 -1 -2 -1 0 1 0 0 1 0 1 0 1 0 0 1

G:=sub<GL(6,Integers())| [1,0,0,0,0,-1,0,1,2,1,-1,-1,0,0,0,1,-1,0,0,0,1,0,0,-1,0,0,2,1,-1,-1,0,0,1,2,-1,-1],[0,1,0,0,-1,0,-1,-1,0,0,0,1,0,0,0,1,-1,0,0,0,-1,-1,0,1,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,1,0,0,-1,0,1,0,0,0,0,-1,1,2,1,0,-1,-1,0,0,0,1,-1,0,2,1,0,0,-1,-1,1,2,0,0,-1,-1],[0,0,-1,-1,0,1,0,0,-1,-1,1,0,-1,-1,-1,0,0,1,0,1,0,-1,0,0,0,0,-1,-2,1,0,0,0,-2,-1,0,1] >;

He3⋊C8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_8
% in TeX

G:=Group("He3:C8");
// GroupNames label

G:=SmallGroup(216,86);
// by ID

G=gap.SmallGroup(216,86);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,12,31,579,681,543,1684,3130,1456,652,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=c,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a*b^-1*c>;
// generators/relations

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