C₅×#73073 #14510797

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73073

Splittings

Fitting

C₄×C₅×#41 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₄⋊S₄
commutator	#41⋊C₃	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₄×C₅×#41	D₃
socle	C₂×C₅	(C₂×S₄)⋊C₄
prime radical	C₅×#5923	C₂
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₄²×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅

322 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236495);