C₅×#67452 #14277666

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#67452

Splittings

Hall	C₃×C₅ ⋊ #1324
Sylow	C₃ ⋊ C₅×#1324

Subgroups

	subgroup	G/H
center	C₂²×C₅	#806
commutator	C₂×C₃×C₄	C₂×C₅×C₈
Frattini	C₂×C₄⋊C₄	C₂×C₅×D₃
Fitting	C₃×C₅×#324	C₂
socle	C₂²×C₃×C₅	#99
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₂×C₃×C₄⋊C₄	C₂²×C₅
hypercenter	C₅×#324	D₃
hypocenter	C₃	C₅×#1324
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄ ◁ C₅×#324

236 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 3364);