C₂²×((C₃⋊D₃)⋊C₃)⋊C₂ #88898

order	432
isoclinism	isoclinic class of ((C₃⋊D₃)⋊C₃)⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂²×((C₃⋊D₃)⋊C₃)⋊C₂
unambiguous semidirect product	((C₃⋊D₃)⋊C₃)⋊C₂³
unambiguous semidirect product	#272⋊C₂³

Fitting	C₂²×H(3,𝔽₃) ⋊ C₂²
derived, Sylow	H(3,𝔽₃) ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂⁴
Frattini	C₃	C₂²×D₃²
Fitting	C₂²×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	D₃²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²

501 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 4 generators on 13 points:

(1,2)(3,4)(6,12,10,8,9,13)(7,11);(5,6,8)(7,9,12,11,10,13);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 545);