C₅×#8758 #11931478

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#8758

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#381
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

394 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 110 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)(8,14,16,11)(9,17)(10,13)(12,18)(15,19,21,20)(22,23,25,29,26,28,32,35)(24,27,30,34,31,33,36,37);(6,8)(7,11)(9,15)(10,16)(12,19)(13,14)(17,20)(18,21)(22,24,26,31)(23,27,28,33)(25,30,32,36)(29,34,35,37);(6,9)(7,12,13,17)(8,15)(10,18)(11,19,14,20)(16,21)(23,28)(25,32)(27,33)(30,36)

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, 1746);