C₅×#11892 #11934612

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11892

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

396 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,19,25,21,27,20,26)(16,28,31,38,33,43,32,42)(17,34,18,37,36,24,35,23)(22,29,39,30,41,45,40,44);(6,8)(7,10)(9,13)(11,12)(14,16,21,33)(15,22,27,41)(17,30,36,44)(18,29,35,45)(19,32,20,31)(23,28,37,43)(24,38,34,42)(25,40,26,39);(8,13)(10,12)(14,17,20,35,21,36,19,18)(15,23,25,34,27,37,26,24)(16,29,32,44,33,45,31,30)(22,38,39,43,41,42,40,28)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4880);