C₅×#11991 #11934711

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11991

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1551
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#540	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#405 ◁ G

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

normal subgroups only

This group has 280 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)(6,7,9,15,22,12,21,25,11,16,24,31,10,14,23,30)(8,17,19,32,34,28,35,36,18,29,33,37,20,13,26,27);(7,12)(9,21)(10,22)(13,17)(14,16)(15,31)(19,35)(20,34)(23,24)(26,33)(27,36)(28,29);(6,8)(7,13,16,28)(9,19)(10,20)(11,18)(12,17,14,29)(15,27,31,36)(21,35)(22,34)(23,26)(24,33)(25,32,30,37)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4979);