C₅×#13437 #11936157

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13437

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#535	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

422 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 86 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)(6,7,12,17)(8,18,22,35)(9,13,11,15)(10,24,25,16)(14,29,31,21)(19,33,20,34)(23,32,26,28)(27,36,30,37);(7,13)(8,19)(9,11)(10,25)(15,17)(16,32)(20,22)(21,36)(24,28)(27,30)(29,37)(33,34);(6,8,9,20,12,22,11,19)(7,14,15,30,17,31,13,27)(10,21,23,37,25,29,26,36)(16,18,32,33,24,35,28,34)

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, 6425);