C₅×#12873 #11935593

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12873

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1823
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#131 ◁ G

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

normal subgroups only

This group has 380 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)(8,17,19,12)(9,21)(10,25)(11,14)(13,18,30,35)(15,23)(16,27)(20,29,26,33)(22,31,36,28)(24,34)(32,37);(6,8)(7,12)(9,18)(10,20)(11,19)(13,21)(14,17)(15,28)(16,29)(22,23)(24,35)(25,33)(26,27)(30,34)(31,32)(36,37);(6,8)(7,13,14,30)(9,22)(10,26)(11,19)(12,21,17,34)(15,33,32,29)(16,28,25,31)(18,23)(20,27)(24,36)(35,37)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5861);