C₅×#11540 #11934260

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11540

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	#1467
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#405 ◁ G

262 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 102 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)(6,7,10,16,11,15,26,32,12,17,24,34,9,13,23,28)(8,18,21,35,22,31,36,33,19,27,25,37,20,14,29,30);(7,13,17,15)(8,19)(10,24)(14,18,31,27)(16,28,34,32)(20,22)(23,26)(30,35,33,37);(6,8,11,22,12,19,9,20)(7,14,13,27,17,31,15,18)(10,25,26,29,24,21,23,36)(16,33,28,35,34,30,32,37)

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, 4528);