C₅×#13432 #11936152

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#13432

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#3199	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

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

normal subgroups only

This group has 440 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)(8,17)(9,12)(10,23)(11,16)(13,20)(14,21)(15,24)(18,31)(19,34)(22,29)(25,35)(26,36)(27,37)(28,33)(30,32);(7,12)(8,18)(9,21)(13,25)(14,16)(15,26)(17,30)(22,36)(23,29)(27,28)(31,34)(35,37);(6,8,9,19,11,18,21,32)(7,13,14,27,16,28,12,25)(10,20,22,35,24,33,36,37)(15,17,29,30,23,31,26,34)

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, 6420);