C₅×#12728 #11935448

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12728

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1782
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×D₄	C₂³×C₅
prime radical	C₅×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×#518 ◁ G

714 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,16,19,29)(9,20)(10,23,25,14)(11,15)(12,18)(13,22)(17,31,33,30)(21,28,37,35)(24,27)(26,36)(32,34);(6,8,11,19)(7,12,15,27)(9,17,22,33)(10,24,25,18)(13,26,20,34)(14,29,23,16)(21,36,37,32)(28,31,35,30);(6,9)(7,13,15,20)(8,17)(10,21)(11,22)(12,26,27,34)(14,28,23,35)(16,30,29,31)(18,32)(19,33)(24,36)(25,37)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5716);