C₅×#13051 #11935771

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13051

Subgroups

	subgroup	G/H
center	C₄×C₅	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1889
prime radical	C₅×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 37 points:

(1,2,3,4,5)(6,7,11,16,9,14,18,27)(8,17,21,33,19,12,26,30)(10,15,24,32,22,13,25,31)(20,28,34,36,23,29,35,37);(6,8,18,26,9,19,11,21)(7,12,27,33,14,17,16,30)(10,23,24,35,22,20,25,34)(13,28,31,36,15,29,32,37);(7,13)(10,22)(12,28)(14,15)(16,32)(17,29)(20,23)(24,25)(27,31)(30,37)(33,36)(34,35)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6039);