C₅×#10388 #11933108

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10388

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

594 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 90 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)(12,13)(14,15,19,22)(16,23)(17,21,18,27)(20,24)(25,29)(26,28);(6,8)(7,10)(9,12)(11,13)(16,24,25,26)(20,23,28,29);(10,13)(14,16)(15,20)(17,24)(18,26)(19,25)(21,29)(22,28)(23,27)

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, 3376);