C₅×#13350 #11936070

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#13350

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₅
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

738 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 94 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)(9,14)(10,13)(11,16)(12,18)(15,23)(19,30)(20,32)(21,27)(22,29)(24,37)(25,33)(26,34)(28,36)(31,35);(6,8)(7,12)(9,18)(10,19)(11,20)(13,26)(14,17)(15,24)(16,25)(21,34)(22,33)(23,35)(27,30)(28,31)(29,32)(36,37);(7,13)(10,23)(12,24)(14,27)(15,16)(17,30)(18,33)(19,35)(21,36)(25,26)(28,29)(31,32)

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, 6338);