C₅×#13355 #11936075

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#13355

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3027	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

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

normal subgroups only

This group has 704 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)(6,7,11,16)(8,17,20,30)(9,21,24,14)(10,13,25,15)(12,22,27,18)(19,32,35,33)(23,31,37,29)(26,34,28,36);(6,8)(7,12)(9,22)(10,19)(11,20)(13,28)(14,30)(15,26)(16,27)(17,21)(18,24)(23,36)(25,35)(29,33)(31,32)(34,37);(7,13)(10,25)(12,26)(14,29)(15,16)(17,32)(18,22)(19,35)(21,31)(23,37)(27,28)(30,33)

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, 6343);