C₅×#85566 #14511304

order	1920
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#85566

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	#1020
commutator, hypocenter	#997	C₄×C₅
Frattini	C₄	C₅×#1020
Fitting	C₅×#585	D₃
socle	C₂×C₅	#5936
prime radical	C₅×#5950	C₂
cosocle	#5950	C₂×C₅
LCS	G ▷ #997
UCS	C₁ ◁ C₄×C₅

332 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5)(6,13,28,34,11,26,36,23)(7,16,32,10,15,29,37,21)(8,22,31,14,18,33,19,27)(9,25,30,12,20,35,17,24);(9,10,19)(13,14,25)(16,17,28)(20,21,31)(22,23,32)(26,27,35)(29,30,36)(33,34,37)

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237002);