C₅×#29659 #11952379

order	1280
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#29659

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

974 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 37 points:

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

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, 22647);