C₅×#69090 #14315753

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

Aliases

direct product

C₅×#69090

Splittings

Fitting	C₃×C₅×#347 ⋊ C₂
Hall	C₃×C₅ ⋊ #2180
Sylow	C₃ ⋊ C₅×#2180

Subgroups

	subgroup	G/H
center	C₂×C₅	#4711
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#347	C₂
socle	C₂×C₃×C₅	#442
prime radical	C₅×#904	C₂²
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#347	D₃
hypocenter	C₃	C₅×#2180
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#97 ◁ C₅×#347

284 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,8)(9,12,14,13,15,26,31,28)(10,17,19,25,20,35,29,18)(11,22,30,27,23,39,24,40)(16,32,37,38,33,21,34,36);(7,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,23)(17,32)(18,36)(19,37)(20,33)(21,35)(24,31)(25,38)(26,39)(28,40)(29,34);(7,8)(9,10)(11,21,23,32)(12,25)(13,17)(14,29)(15,20)(16,22,33,39)(18,26)(19,31)(24,36,30,38)(27,37,40,34)(28,35)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 41451);