C₅×#7207 #11929927

order	1280
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7207

Subgroups

	subgroup	G/H
center	C₅×C₈	#99
commutator	C₂³	C₄×C₅×C₈
Frattini, cosocle	C₂×#131	C₂²×C₅
socle	C₂×C₅	#1282
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ 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)(6,7,10,16,12,19,32,43,13,18,11,17,31,42,33,44)(8,20,22,45,24,48,53,67,25,47,23,46,52,66,54,68)(9,26,27,41,28,55,56,65,29,15,30,38,57,39,58,63)(14,34,35,59,37,60,61,21,36,49,40,50,62,69,64,51);(6,8,11,23,12,24,33,54,13,25,10,22,31,52,32,53)(7,14,17,35,19,37,44,61,18,36,16,40,42,62,43,64)(9,21,27,49,28,50,56,69,29,51,30,34,57,59,58,60)(15,20,38,46,39,48,63,68,26,47,41,45,55,66,65,67)

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, 195);