C₅×#73293 #14511017

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73293

Splittings

Fitting, Hall	C₅×#2840 ⋊ C₃
derived, Sylow	#2840 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂³×C₅	C₂²⋊A₄
commutator, cosocle, hypocenter	#2840	C₃×C₅
Frattini	C₂³	C₅×C₂²⋊A₄
Fitting	C₅×#2840	C₃
LCS	G ▷ #2840
UCS	C₁ ◁ C₂³×C₅

370 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 160

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

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236715);