C₅⋊((C₂²⋊C₉)⋊C₂) #67111

order	360
isoclinism	isoclinic class of C₅⋊((C₂²⋊C₉)⋊C₂)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₅⋊((C₂²⋊C₉)⋊C₂)
unambiguous semidirect product	(C₂²⋊C₉)⋊D₅

derived	C₅×C₂²⋊C₉ ⋊ C₂
Sylow	C₅ ⋊ (C₂²⋊C₉)⋊C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅×C₂²⋊C₉	C₂
Frattini	C₃	C₅⋊S₄
Fitting, socle	C₂²×C₃×C₅	D₃
LCS	G ▷ C₅×C₂²⋊C₉
UCS	C₁

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,4,3)(6,7)(8,9)(11,13)(12,16)(14,15)(17,18);(2,3,4)(5,6,8,9,7)(10,18,13,16,15,14,12,11,17)

Polycyclic presentation:

2,3,5,3,2^2|b^3=d^2,b^a=b^2d,c^a=c^4,d^a=d^2,e^a=f,f^a=e,e^b=f,f^b=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 41);