C₅×#11805 #11934525

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11805

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₂×C₅×#121	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁴×C₅ ◁ C₂²×C₅×C₄⋊C₄ ◁ G

448 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 98 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)(14,15,17,21,18,20,25,28)(16,22,24,29,23,19,26,27);(7,10)(8,13)(11,12)(14,16)(15,19)(17,24)(18,23)(20,22)(21,29)(25,26)(27,28);(7,11)(10,12)(16,23)(19,22)(24,26)(27,29)

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, 4793);