C₅×#23897 #11946617

order	1280
nilpotency class	3
isoclinism	isoclinic class of #23897
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₅×#23897

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ G

1952 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 282 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 — 4 generators on 21 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#23897, AGL(1,𝔽₅)×#23897, #72243091, …
maximal subgroup of	D₅×#23897, #72243135, C₅×#511790
maximal normal subgroup of	D₅×#23897, #72243135, C₅×#511790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 16885);