C₅×#11838 #11934558

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11838

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1514
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×M₁₆ ◁ G

280 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 37 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4826);