C₅×#12313 #11935033

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12310
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12313

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 356 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)(6,7,12,8)(9,19,22,35)(10,24,29,25)(11,18,27,15)(13,30,16,21)(14,33,17,28)(20,34,23,36)(26,31,37,32);(6,8,12,7)(9,16,22,13)(10,25,29,24)(11,15,27,18)(14,28,17,33)(19,26,35,37)(20,31,23,32)(21,36,30,34);(6,9)(7,13)(8,16)(10,26)(11,20)(12,22)(14,34)(15,31)(17,36)(18,32)(19,24)(21,28)(23,27)(25,35)(29,37)(30,33)

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, 5301);