C₅×#12969 #11935689

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12969

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	#1867
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

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

normal subgroups only

This group has 416 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)(6,7,11,16,12,17,10,25)(8,18,21,32,22,35,26,29)(9,15,23,14,24,33,27,34)(13,28,30,19,31,36,37,20);(6,8,11,21,12,22,10,26)(7,13,16,30,17,31,25,37)(9,20,23,28,24,19,27,36)(14,32,33,35,34,29,15,18);(6,9)(7,14)(8,19)(10,27)(11,23)(12,24)(13,29)(15,25)(16,33)(17,34)(18,30)(20,22)(21,36)(26,28)(31,32)(35,37)

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, 5957);