C₅×#7433 #11930153

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7433

Subgroups

	subgroup	G/H
center	C₄×C₅	#353
commutator	C₂×M₁₆	C₂×C₄×C₅
Frattini, cosocle	#407	C₂²×C₅
socle	C₂×C₅	#1352
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#407 ◁ G

182 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 37 points:

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

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, 421);