C₅×#12730 #11935450

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×#12730

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1782
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×#519 ◁ G

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

normal subgroups only

This group has 666 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 5718);