C₅×#12810 #11935530

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12810

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1799
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131 ◁ G

412 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 126 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)(6,7,11,15,12,17,9,23)(8,18,22,28,19,33,20,13)(10,16,24,30,25,14,26,31)(21,32,35,27,34,37,36,29);(6,8,9,20,12,19,11,22)(7,13,15,18,17,28,23,33)(10,21,24,35,25,34,26,36)(14,29,30,37,16,27,31,32);(7,14)(8,19)(10,25)(13,27)(15,30)(16,17)(18,32)(20,22)(23,31)(24,26)(28,29)(33,37)

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, 5798);