C₅×#11842 #11934562

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11842

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

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

normal subgroups only

This group has 390 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4830);