#88390⋊C₃ #13046947

order	1296
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, hypercenter	C₂	((C₂²⋊C₂₇)⋊C₂)⋊C₃
commutator, cosocle, hypocenter	Q₈⋊C₂₇	C₂×C₃
Frattini	C₂×C₉	C₃×S₄
Fitting	C₃×Q₈×C₉	D₃
socle	C₂×C₃	C₃×(C₂²⋊C₉)⋊C₂
prime radical	C₂×C₃²	(C₂²⋊C₉)⋊C₂
LCS	G ▷ Q₈⋊C₂₇
UCS	C₁ ◁ C₂

66 conjugacy classes of subgroups.

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 432

Polycyclic presentation:

2,3^4,2^3|a^2=h,c^3=d,d^3=e,f^2=h,g^2=h,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^a=fg,g^a=gh,c^b=ce^2,f^c=g,g^c=fgh,g^f=gh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 665);