C₅×#7046 #11929766

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7046

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#95
commutator	C₂²×C₄	C₄²×C₅
Frattini, cosocle	C₂²×C₄²	C₂²×C₅
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

364 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,9,12,10,13,8,11)(14,15)(16,20)(17,19)(18,21)(22,23)(24,30,31,27)(25,29)(26,28)(32,36,37,34)(33,35);(7,11,13,12)(14,16)(15,18,19,20)(17,21)(22,24,25,32,26,31,33,37)(23,27,29,34,28,30,35,36)

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, 34);