C₅×#87517 #14511451

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#87517

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×#1020
commutator, cosocle, hypocenter	#997	C₂²×C₅
Frattini	C₂	C₂×C₅×#1020
Fitting	C₅×#585	D₃
prime radical	C₅×#5950	C₂
hypercenter	C₄×C₅	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅

520 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 32 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)(6,8,11,19)(7,12,15,16)(9,18,20,32,10,37,21,34,27,17,23,31)(13,25,22,29,14,33,26,30,36,24,28,35);(6,22,21,28)(7,27,26,10)(8,33,32,25)(9,14,11,36)(12,31,29,34)(13,23,15,20)(16,37,24,18)(17,30,19,35)

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237149);