C₅×#7208 #11929928

order	1280
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7208

Subgroups

	subgroup	G/H
center	C₅×C₈	#99
commutator	C₂×C₄	C₄×C₅×C₈
Frattini, cosocle	C₂×#131	C₂²×C₅
socle	C₂×C₅	#1282
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131 ◁ G

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5)(6,7,10,16,12,18,31,41,13,17,11,32,30,42,33,58)(8,19,21,43,22,46,50,66,23,45,28,44,51,65,55,67)(9,24,25,39,26,52,53,64,27,15,29,38,54,40,56,63)(14,34,35,48,37,60,61,49,36,47,57,59,62,68,69,20);(6,8,11,21,12,22,33,50,13,23,10,28,30,51,31,55)(7,14,16,35,18,37,41,61,17,36,32,57,42,62,58,69)(9,20,25,47,26,48,53,68,27,49,29,34,54,59,56,60)(15,19,39,44,40,46,64,67,24,45,38,43,52,65,63,66)

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, 196);