C₅×#8130 #11930850

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#8130

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#95
commutator	C₂³	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄²×C₅ ◁ G

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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)(6,7,10,13,11,14,9,18)(8,15,16,20,17,12,19,21)(22,23,25,30,27,29,26,34)(24,28,32,36,31,35,33,37);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,15)(18,21)(22,24,26,32,27,31,25,33)(23,28,30,37,29,35,34,36);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,15,21)(23,29)(28,35)(30,34)(36,37)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1118);