C₅×#11916 #11934636

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11916

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1533
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#97	C₂³×C₅
prime radical	C₂³×C₅	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×M₁₆ ◁ G

274 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 90 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)(6,7,9,15,10,16,23,34,11,13,22,30,25,14,31,32)(8,17,18,35,19,12,26,28,20,29,36,33,24,27,21,37);(6,8,11,20)(7,12,13,27)(9,21,22,26)(10,24,25,19)(14,29,16,17)(15,33,30,35)(18,23,36,31)(28,32,37,34);(7,13)(9,22)(12,27)(14,16)(17,29)(18,36)(21,26)(23,31)

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, 4904);