C₅×#85385 #14511123

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85385

Splittings

Fitting

C₅×#548 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#548	D₃
hypercenter	C₅×C₂²⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

556 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,3,4,5)(6,7)(8,9)(10,18,21)(11,16,25,15,22,24)(12,13,20)(17,23);(7,9)(10,11,12,15)(13,16,18,22)(14,17,19,23)(20,24,21,25);(7,9)(13,19)(14,18)(16,23)(17,22)(20,21)(24,25)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236821);