C₂²⋊C₂₇ #1076

order	108
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₉×S₄ (order 216) · Inn(G) = A₄ · Out(G) = C₂×C₉
holomorph	G ⋊ Aut(G) = #72259339
homology	Abelianization (H₁) = C₂₇ · Schur multiplier (H₂) = C₂

Aliases

C₂²⋊C₂₇

C₂² ⋊ C₂₇

	subgroup	G/H
center, Frattini, hypercenter	C₉	A₄
commutator, hypocenter	C₂²	C₂₇
Fitting, cosocle	C₂²×C₉	C₃
socle, prime radical	C₂²×C₃	C₉
LCS	G ▷ C₂²
UCS	C₁ ◁ C₉

11 conjugacy classes of subgroups.

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 31 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 3);