C₅×#7312 #11930032

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7312

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1304
commutator	C₂⁴	C₄²×C₅
Frattini, cosocle	C₂²×#41	C₂²×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×#41 ◁ G

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7)(8,16,18,27)(9,20,21,13)(10,23,24,14)(11,15)(12,17)(19,34)(22,28)(25,31,36,37)(26,32)(29,35)(30,33);(6,8,10,19,11,18,24,33)(7,12,14,25,15,26,23,36)(9,17,22,31,21,32,35,37)(13,27,28,34,20,16,29,30)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 300);