C₅×#11922 #11934642

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11922

Subgroups

	subgroup	G/H
center	C₄×C₅	#331
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	C₂×#331
prime radical	C₅×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 400 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)(6,7,10,14,23,15,33,31,11,16,25,32,9,13,22,29)(8,17,19,34,18,12,24,27,21,26,36,30,20,28,35,37);(6,8)(7,12)(9,18)(10,24)(11,21)(13,26)(14,30)(15,17)(16,28)(19,33)(20,23)(22,36)(25,35)(27,31)(29,37)(32,34);(6,9,11,23)(7,13,16,15)(8,18,21,20)(10,22,25,33)(12,26,28,17)(14,29,32,31)(19,24,36,35)(27,30,37,34)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4910);