C₅×#12968 #11935688

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12968

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

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

normal subgroups only

This group has 424 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)(6,7,11,16,12,17,10,24)(8,18,20,28,21,35,25,34)(9,15,22,14,23,31,26,32)(13,27,29,37,30,36,33,19);(6,8,11,20,12,21,10,25)(7,13,16,29,17,30,24,33)(9,19,22,27,23,37,26,36)(14,28,31,35,32,34,15,18);(6,9)(7,14)(8,19)(10,26)(11,22)(12,23)(13,28)(15,24)(16,31)(17,32)(18,33)(20,27)(21,37)(25,36)(29,35)(30,34)

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, 5956);