C₅×#85666 #14511404

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#85666

Splittings

Fitting, Hall	C₅×#3573 ⋊ C₃
Sylow	#3573 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₈	C₂²⋊A₄
commutator, hypocenter	H(5,𝔽₂)	C₃×C₄×C₅
Frattini	C₄	C₂×C₅×C₂²⋊A₄
Fitting	C₅×#3573	C₃
socle	C₂×C₅	C₄×C₂²⋊A₄
prime radical	C₅×#5950	C₂
cosocle	#585	C₂×C₃×C₅
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₅×C₈

558 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,8,13,12,17,21,28)(9,29,20,25,23,15,34,36,11,14,32,27,24,30,19,37,22,16,18,35,10,31,33,26);(9,24)(10,11)(14,31)(15,16)(18,34)(19,20)(25,37)(26,27);(6,23)(7,30)(8,33)(9,11)(10,24)(12,22)(13,36)(14,16)(15,31)(17,29)(18,20)(19,34)(21,32)(25,27)(26,37)(28,35)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237102);