C₅×#11993 #11934713

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11992
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11993

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1555
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×M₁₆ ◁ G

360 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,9,15,10,14,23,30,11,13,22,29,25,16,33,32)(8,17,18,34,19,26,36,37,20,28,35,31,24,12,21,27);(6,8,11,20)(7,12,13,26)(9,21,22,36)(10,24,25,19)(14,28,16,17)(15,31,29,34)(18,23,35,33)(27,30,37,32);(7,13)(9,22)(12,26)(14,16)(17,28)(18,35)(21,36)(23,33)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4981);