C₅×#11311 #11934031

order	1280
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#11311

Subgroups

	subgroup	G/H
center	C₄×C₅	#396
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂×C₅	#1443
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

532 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 202 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)(8,18)(9,22,26,14)(10,16)(11,17)(12,15)(13,20)(19,30,34,35)(21,33)(23,28,36,37)(24,31,25,32)(27,29);(6,8,12,20,11,21,10,27)(7,13,15,33,17,29,16,18)(9,23,24,19,26,36,25,34)(14,30,32,37,22,35,31,28);(6,9,10,24,11,26,12,25)(7,14,15,31,17,22,16,32)(8,19,20,23,21,34,27,36)(13,28,18,30,29,37,33,35)

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, 4299);