C₅×#12315 #11935035

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12315

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3291	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

362 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 78 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)(6,7,11,15)(8,16,19,35)(9,21,24,31)(10,25,26,14)(12,27,29,18)(13,22,33,37)(17,34,20,32)(23,30,36,28);(9,22)(10,26)(13,31)(14,25)(16,34)(17,20)(18,36)(21,33)(23,27)(24,37)(28,30)(32,35);(6,8)(7,12)(9,23)(10,17)(11,19)(13,32)(14,28)(15,29)(16,21)(18,37)(20,26)(22,27)(24,36)(25,30)(31,35)(33,34)

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, 5303);