C₅×#12465 #11935185

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12465

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ G

196 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 70 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)(7,13)(8,18)(9,14)(10,25)(11,24)(16,36)(17,26)(19,20)(21,23)(22,30)(27,32)(28,29)(31,37)(33,34);(6,7,14,29,11,16,10,26,12,17,25,36,24,28,9,13)(8,15,30,32,20,33,19,31,22,35,18,37,21,34,23,27);(6,8)(7,15)(9,23)(10,19)(11,20)(12,22)(13,27)(14,30)(16,33)(17,35)(18,25)(21,24)(26,31)(28,34)(29,32)(36,37)

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, 5453);