C₅×#7226 #11929946

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7226

Subgroups

	subgroup	G/H
center	C₄×C₅	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₄×M₁₆	C₂²×C₅
socle	C₂×C₅	#1288
prime radical	C₂³×C₅	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ G

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 214);