C₅×#85322 #14511060

order	1920
isoclinism	isoclinic class of #5386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#85322

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₂×C₄×C₅×Q₈	D₃
socle	C₂²×C₅	C₂²×S₄
prime radical	C₂²×C₅×SL(2,𝔽₃)	C₂²
hypercenter	C₂²×C₄×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅

424 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 84 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)(15,25,29)(16,23,18)(19,28,21);(6,8,9,12)(7,10,11,13);(6,7)(8,10)(9,11)(12,13)(14,15,18,21)(16,22,24,20)(17,25,27,19)(23,29,26,28)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236758);