C₅×#7286 #11930006

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7286

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂×C₄×Q₈	C₂²×C₅
socle	C₂²×C₅	#342
prime radical	C₂³×C₅	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×Q₈ ◁ G

250 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 2 generators on 37 points:

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

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, 274);