C₅×#7145 #11929865

order	1280
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7145

Subgroups

	subgroup	G/H
center	C₄²×C₅	C₄²
commutator	C₄	C₄×C₅×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ G

222 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 126 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)(8,15,17,21)(9,13)(10,19)(11,14)(12,16,20,18)(22,23,24,27,25,28,30,33,26,29,31,34,32,35,36,37);(6,8,10,18,11,17,9,16)(7,12,13,15,14,20,19,21)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 133);