C₅×#12367 #11935087

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12367

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 78 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)(8,16,18,35)(9,20)(10,23)(12,27,29,37)(13,31)(14,26)(17,22,19,34)(21,33)(24,25)(28,32,30,36);(6,7,11,15)(8,12,18,29)(9,21,25,31)(10,26,23,14)(13,20,33,24)(16,32,35,36)(17,30,19,28)(22,27,34,37);(6,8)(7,12)(9,22)(10,17)(11,18)(13,32)(14,28)(15,29)(16,24)(19,23)(20,35)(21,27)(25,34)(26,30)(31,37)(33,36)

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, 5355);