C₂²⋊(C₂²⋊C₂₇) #88579

order	432
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂²⋊(C₂²⋊C₂₇)

C₂⁴ ⋊ C₂₇

	subgroup	G/H
center, Frattini, hypercenter	C₉	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₂₇
Fitting, cosocle	C₂⁴×C₉	C₃
socle, prime radical	C₂⁴×C₃	C₉
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₉

88 conjugacy classes of subgroups.

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 35 points:

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

Polycyclic presentation:

3^3,2^4|a^3=b,b^3=c,d^a=e,e^a=de,f^a=g,g^a=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 226);