C₅×#69085 #14315691

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

Aliases

direct product

C₅×#69085

Splittings

Fitting	C₃×C₅×#346 ⋊ C₂
Hall	C₃×C₅ ⋊ #2179
Sylow	C₃ ⋊ C₅×#2179

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×D₂₄
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#346	C₂
socle	C₂×C₃×C₅	C₄×D₈
prime radical	C₅×#4904	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#346	D₃
hypocenter	C₃	C₅×#2179
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₈ ◁ C₅×#346

288 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 114 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)(9,12,14,27,15,28,31,13)(10,17,19,29,21,18,20,36)(11,23,24,32,26,40,25,30)(16,33,34,22,35,37,38,39);(7,8)(9,11)(10,16)(12,23)(13,30)(14,24)(15,26)(17,33)(18,37)(19,34)(20,38)(21,35)(22,29)(25,31)(27,32)(28,40)(36,39);(7,8)(9,10,14,19,15,21,31,20)(11,22,24,37,26,39,25,33)(12,18,27,36,28,17,13,29)(16,32,34,40,35,30,38,23)

Conjugacy classes & character table

360 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 41389);