C₅×#12727 #11935447

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12727

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1782
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×D₄	C₂³×C₅
prime radical	C₅×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×#518 ◁ G

810 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5715);