C₂₇⋊C₈ #6164

order	216
isoclinism	isoclinic class of D₂₇
automorphism	Aut(G) = C₂²×AGL(1,ℤ/27) (order 1944) · Inn(G) = D₂₇ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₂₇⋊C₈

Splittings

derived, Sylow

C₂₇ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₂₇
commutator, hypocenter	C₂₇	C₈
Frattini	C₄×C₉	D₃
Fitting, cosocle	C₄×C₂₇	C₂
socle, prime radical	C₂×C₃	Dic₉
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 2 generators on 35 points:

(2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24)(17,23)(21,22)(26,27)(28,29,30,32,31,33,34,35);(1,20,14,21,25,23,3,2,24,4,27,18,10,7,6,11,8,26,12,16,5,9,17,15,22,19,13)

Polycyclic presentation:

2^3,3^3|a^2=b,b^2=c,d^3=e^2f,e^3=f^2,d^a=d^2e,e^a=e^2f,f^a=f^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₂₇, #88359, C₂×C₂₇⋊C₈, …
maximal normal subgroup of	C₈×D₂₇, #88359, C₂×C₂₇⋊C₈, …
maximal quotient of	C₂₇⋊C₁₆, C₂×C₂₇⋊C₈, C₈₁⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 1);

C27⋊C8 #6164