C₅×#81791 #14458173

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#81791

Splittings

Fitting	C₃×C₅×#493 ⋊ C₂
Hall	C₃×C₅ ⋊ #3292
Sylow	C₃ ⋊ C₅×#3292

Subgroups

	subgroup	G/H
center	C₂×C₅	#5717
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#493	C₂
socle	C₂×C₃×C₅	D₄²
hypercenter	C₅×#493	D₃
hypocenter	C₃	C₅×#3292
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#493

952 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,4,5)(6,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21);(7,8)(9,10)(11,16)(12,15)(13,21)(14,17)(18,19)(20,24)(22,23);(7,8)(9,10,14,17)(11,18,20,23)(12,21,13,15)(16,22,24,19);(10,15)(12,13)(16,23)(17,21)(18,24)(19,22)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 183871);