C₅×C₂²⋊#997 #14511484

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product	C₅×C₂²⋊#997
unambiguous semidirect product	C₂²⋊(C₅×#997)
unambiguous semidirect product	(C₂²×C₅)⋊#997

Fitting, Hall	C₂²×C₅×H(5,𝔽₂) ⋊ C₃
derived, Sylow	C₂²×H(5,𝔽₂) ⋊ C₃×C₅

	subgroup	G/H
center, hypercenter	C₂×C₅	C₂²⋊(C₂²⋊A₄)
commutator, cosocle, hypocenter	C₂²×H(5,𝔽₂)	C₃×C₅
Frattini	C₂	C₅×C₂²⋊(C₂²⋊A₄)
Fitting	C₂²×C₅×H(5,𝔽₂)	C₃
socle	C₂³×C₅	C₂²⋊A₄
LCS	G ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₅

2390 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 2390 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 4 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237182);