C₅×#85332 #14511070

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#85332

Splittings

Fitting

C₂×C₄×C₅×Q₈ ⋊ 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	C₂×C₄×C₅×Q₈	D₃
hypercenter	C₂²×C₄×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅

482 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(7,11)(10,13)(14,24,26)(15,20,29,19,25,28)(16,23,18)(21,22);(6,8,9,12)(7,10,11,13)(15,19)(20,25)(21,22)(28,29);(6,7)(8,10)(9,11)(12,13)(14,15)(16,22)(17,25)(18,19)(20,27)(21,24)(23,29)(26,28)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236768);