C₅×#13490 #11936210

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13490

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2073
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

292 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 94 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)(6,7,12,24,9,13,11,16)(8,14,18,30,19,26,23,34)(10,15,25,31,20,27,21,33)(17,29,22,32,35,28,36,37);(7,13)(8,17)(10,21)(11,12)(14,28)(15,31)(18,36)(19,35)(20,25)(22,23)(26,29)(27,33)(30,32)(34,37);(6,8)(7,14,13,26)(9,19)(10,22)(11,23)(12,18)(15,32,27,37)(16,34,24,30)(17,21)(20,36)(25,35)(28,33,29,31)

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, 6478);