C₅×#7095 #11929815

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#7095

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#331
commutator	C₄×C₈	C₂×C₄×C₅
Frattini, cosocle	C₈²	C₂²×C₅
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 1280

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 83);