C₅×#13578 #11936298

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13578
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#13578

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×#353
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	#124	C₂³×C₅
prime radical	C₅×#3153	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#124 ◁ G

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

normal subgroups only

This group has 420 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)(8,16,19,31)(9,20,21,36)(10,23,25,14)(11,15)(12,18,27,34)(13,22)(17,29,24,32)(26,33,28,35)(30,37);(6,8)(7,12)(9,18)(10,24)(11,17)(13,29)(14,27)(15,26)(16,30)(19,25)(20,32)(21,33)(22,34)(23,28)(31,36)(35,37);(8,17)(12,26)(16,29)(18,33)(19,24)(27,28)(31,32)(34,35)

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, 6566);