C₅×#7092 #11929812

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

Aliases

direct product

C₅×#7092

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#331
commutator	C₄×C₈	C₂×C₄×C₅
Frattini, cosocle	C₈²	C₂²×C₅
prime radical	C₅×#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

152 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 640

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

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

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, 80);