C₅×#7315 #11930035

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7313
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7315

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1304
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂³×Q₈	C₂²×C₅
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂³×C₅×Q₈ ◁ G

432 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 37 points:

(1,2,3,4,5)(6,7,10,15,12,17,25,33)(8,18,20,34,21,28,36,37)(9,16,23,32,11,14,24,30)(13,26,27,19,29,35,31,22);(6,8,11,19,12,21,9,22)(7,13,16,28,17,29,14,18)(10,20,23,26,25,36,24,35)(15,31,30,37,33,27,32,34)

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, 303);