C₅×#69159 #14313527

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

Aliases

direct product

C₅×#69159

Splittings

Fitting	C₃×C₅×#354 ⋊ C₂
Hall	C₃×C₅ ⋊ #2129
Sylow	C₃ ⋊ C₅×#2129

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×#99
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂×D₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#354	C₂
socle	C₂×C₃×C₅	C₂×#99
prime radical	C₂×C₅×#237	C₂²
cosocle	C₂×C₃×D₄	C₂³×C₅
hypercenter	C₅×#354	D₃
hypocenter	C₃	C₅×#2129
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₅×D₄ ◁ C₅×#354

476 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 110 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)(7,8)(9,12,14,25)(10,16,18,33)(11,20,22,23)(13,24,28,40)(15,29,30,31)(17,32,26,39)(19,35,34,37)(21,36,27,38);(6,7,8)(9,11,14,22)(10,15,18,30)(12,23,25,20)(13,27,28,21)(16,31,33,29)(17,34,26,19)(24,36,40,38)(32,35,39,37);(9,10)(11,19,22,34)(12,16)(13,26,28,17)(14,18)(15,21)(20,35,23,37)(24,39,40,32)(25,33)(27,30)(29,36)(31,38)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 39225);