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G = He3⋊C8  order 216 = 23·33

The semidirect product of He3 and C8 acting faithfully

non-abelian, soluble

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 >

C1
9C2
C3
12C3
9C4
9C6
12S3
4C32
27C8
9C12
12C3×S3
He3
9C3⋊C8
He3⋊C2
He3⋊C4
He3⋊C8

Character table of He3⋊C8

 class 123A3B4A4B68A8B8C8D12A12B
 size 192249918272727271818
ρ11111111111111    trivial
ρ21111111-1-1-1-111    linear of order 2
ρ31111-1-11-ii-ii-1-1    linear of order 4
ρ41111-1-11i-ii-i-1-1    linear of order 4
ρ51-111-ii-1ζ8ζ83ζ85ζ87-ii    linear of order 8
ρ61-111-ii-1ζ85ζ87ζ8ζ83-ii    linear of order 8
ρ71-111i-i-1ζ87ζ85ζ83ζ8i-i    linear of order 8
ρ81-111i-i-1ζ83ζ8ζ87ζ85i-i    linear of order 8
ρ96-2-302210000-1-1    orthogonal faithful
ρ106-2-30-2-21000011    symplectic faithful, Schur index 2
ρ1162-302i-2i-10000-ii    complex faithful
ρ1262-30-2i2i-10000i-i    complex faithful
ρ13808-1000000000    orthogonal lifted from F9

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

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

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

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

100000
010000
020121
011012
0-1-10-1-1
-1-10-1-1-1
,
0-10000
1-10000
000-100
001-100
-10-10-1-1
010110
,
011021
102012
001000
000100
-10-1-1-1-1
0-1-10-1-1
,
00-1000
00-1100
-1-1-10-1-2
-1-10-1-2-1
010010
101001

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
 

Export

Subgroup lattice of He3⋊C8 in TeX
Character table of He3⋊C8 in TeX

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