C₅×#68369 #14284481

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

Aliases

direct product

C₅×#68369

Splittings

Fitting	C₃×C₅×#330 ⋊ C₂
Hall	C₃×C₅ ⋊ #1537
Sylow	C₃ ⋊ C₅×#1537

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#330	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#330	D₃
hypocenter	C₃	C₅×#1537
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#330

388 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 960

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

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

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, 10179);