C₅×#8827 #11931547

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#8827

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#384
prime radical	C₂³×C₅	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

306 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7,10,13,11,14,9,19)(8,15,18,20,16,21,17,12)(22,23)(24,30)(25,27)(26,34)(28,32)(29,33)(31,36)(35,37);(6,8,9,17,11,16,10,18)(7,12,13,15,14,20,19,21)(23,27)(28,35)(29,34)(30,36);(8,16)(12,20)(15,21)(17,18)(22,24,26,32,25,31,33,37)(23,28,29,30,27,35,34,36)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1815);