C₅×#11844 #11934564

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₅×#11844

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	C₅×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

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

normal subgroups only

This group has 432 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) = 160

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

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

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, 4832);