C₅×#7188 #11929908

order	1280
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7188

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂³	C₄×C₅×C₈
Frattini, cosocle	C₂×#117	C₂²×C₅
socle	C₂²×C₅	#342
prime radical	C₂⁴×C₅	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

322 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5)(6,7,8,10,9,11,12,13)(14,15)(16,24)(17,28,29,21)(18,22)(19,23)(20,25,32,38)(26,36)(27,37)(30,41,42,34)(31,35)(33,39,43,45)(40,44);(14,16,18,26,19,27,31,40)(15,20,22,33,23,32,35,43)(17,25,30,39,29,38,42,45)(21,24,34,36,28,37,41,44)

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, 176);