C₅×#85345 #14511083

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

solvable

Aliases

direct product

C₅×#85345

Splittings

Fitting

C₄×C₅×#41 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₄×C₅×#41	D₃
socle	C₂²×C₅	C₂²×S₄
hypercenter	C₄²×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅

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

normal subgroups only

This group has 480 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)(6,9,10,8)(7,11,12,13)(14,22,25)(15,26,29)(16,17,24)(19,20,28);(6,8,10,9)(7,11,12,13)(14,15,16,19)(17,20,22,26)(18,21,23,27)(24,28,25,29);(6,7)(8,13)(9,11)(10,12)(15,19)(17,23)(18,22)(20,21)(24,25)(26,27)

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, 236781);