C₅×#11395 #11934115

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#11395

Subgroups

	subgroup	G/H
center	C₄×C₅	#393
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂×C₅	#1446
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

820 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 250 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)(6,7,10,15,11,14,23,30)(8,12,18,26,17,25,22,31)(9,19,21,34,20,13,27,29)(16,28,33,37,32,24,35,36);(6,8)(7,12,14,25)(9,16)(10,22)(11,17)(13,28,19,24)(15,31,30,26)(18,23)(20,32)(21,35)(27,33)(29,36,34,37);(6,9)(7,13,14,19)(8,16)(10,21)(11,20)(12,24,25,28)(15,29,30,34)(17,32)(18,33)(22,35)(23,27)(26,36,31,37)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4383);