C₂²×(C₂²⋊C₉)⋊C₂ #64187

order	288
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₂²×(C₂²⋊C₉)⋊C₂
unambiguous semidirect product	(C₂²⋊C₉)⋊C₂³
unambiguous semidirect product	(C₂×C₂²⋊C₉)⋊C₂²

C₂²⋊C₉ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂³
Frattini	C₃	C₂²×S₄
Fitting, socle	C₂⁴×C₃	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂²

230 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 17 points:

(5,6,8,7)(10,12)(11,15)(13,14)(16,17);(3,4)(6,7,8)(9,17,12,15,14,13,11,10,16);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 835);