((C₃⋊S₄)⋊C₃)⋊C₂ #88876

order	432
isoclinism	isoclinic class of ((C₃⋊S₄)⋊C₃)⋊C₂
automorphism	Aut(G) = ((C₃⋊S₄)⋊C₃)⋊C₂ (order 432) · Inn(G) = G · Out(G) = C₁
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product	((C₃⋊S₄)⋊C₃)⋊C₂
unambiguous semidirect product	#6258⋊C₂
unambiguous semidirect product	((C₃⋊D₃)⋊A₄)⋊C₂

Fitting	C₂²×C₃² ⋊ C₂×D₃
derived	C₂²⋊H(3,𝔽₃) ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²⋊H(3,𝔽₃)	C₂²
Frattini	C₃	D₃×S₄
Fitting	C₂²×C₃²	C₂×D₃
socle	C₂²×C₃	D₃²
LCS	G ▷ C₂²⋊H(3,𝔽₃)
UCS	C₁

134 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 523);