C₅×#10022 #11932742

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10022

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#2631	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

576 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,10,17)(8,12,19,14)(9,13,22,31)(11,18,24,16)(15,25,32,23)(20,29,33,28)(21,36,34,30)(26,37,27,35);(7,12)(8,19)(9,22)(13,30)(14,17)(15,26)(16,28)(18,29)(20,33)(23,25)(27,32)(31,36);(7,13)(8,20)(9,23)(12,26)(14,27)(15,16)(17,31)(18,32)(19,33)(21,34)(22,25)(28,30)(29,36)(35,37)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3010);