C₅×#10005 #11932725

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10005

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3109	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

566 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(7,12)(8,13)(9,10)(14,15)(16,20)(17,19)(18,22)(21,25)(23,28)(24,29)(26,27);(6,7)(8,11)(9,13)(10,12)(15,19)(17,25)(20,28)(21,22)(23,26)(27,29);(6,8,11,7)(9,13,10,12)(14,16,18,24)(15,20,22,29)(17,26,25,23)(19,27,21,28)

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, 2993);