C₅×#10366 #11933086

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#10366

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

402 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,8,10)(9,11,13,12)(14,15)(16,23,25,29)(17,21)(18,27)(19,22)(20,26,28,24);(7,10)(11,12)(16,24,25,26)(20,23,28,29);(7,11)(9,13)(10,12)(14,16)(15,20)(17,24)(18,26)(19,25)(21,29)(22,28)(23,27)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3354);