C₅×#11312 #11934032

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#11312

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#396
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

538 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4300);