C₅×#85329 #14511067

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

solvable

Aliases

direct product

C₅×#85329

Splittings

Fitting

C₂×C₄×C₅×Q₈ ⋊ 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	C₂×C₄×C₅×Q₈	D₃
socle	C₂²×C₅	C₂²×S₄
prime radical	C₅×#5421	C₂
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

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

normal subgroups only

This group has 448 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)(6,8,9,12)(7,11,10,13)(14,22,27)(15,26,29)(16,17,25)(19,20,28);(7,10)(11,13)(15,19)(20,26)(21,24)(28,29);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,24)(18,26)(20,23)(21,22)(25,29)(27,28)

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, 236765);