C₅×#70993 #14316929

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

Aliases

direct product

C₅×#70993

Splittings

Fitting	C₃×C₅×#456 ⋊ C₂
Hall	C₃×C₅ ⋊ #2203
Sylow	C₃ ⋊ C₅×#2203

Subgroups

	subgroup	G/H
center	C₂×C₅	#5032
commutator, cosocle	C₂×C₃×Q₈	C₂³×C₅
Frattini	C₂×Q₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#456	C₂
socle	C₂×C₃×C₅	B(4,𝔽₂)
prime radical	C₅×#5245	C₂
hypercenter	C₅×#456	D₃
hypocenter	C₃	C₅×#2203
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈ ◁ C₅×#456

432 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(7,13)(10,26)(12,28)(14,33)(15,16)(17,34)(19,25)(21,31)(23,27)(29,30)(32,35)(36,37)(39,40);(6,8,11,20)(7,12,16,30)(9,22,24,18)(10,25,26,19)(13,28,15,29)(14,32,21,17)(23,36,27,37)(31,34,33,35)(38,39,40);(6,7)(8,17)(9,21)(10,13)(11,16)(12,18)(14,24)(15,26)(19,34)(20,32)(22,30)(23,31)(25,35)(27,33)(28,36)(29,37)

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42627);