C₅×#12068 #11934788

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12068

Subgroups

	subgroup	G/H
center	C₄×C₅	#442
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	#1580
prime radical	C₅×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#346 ◁ G

204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#12068, AGL(1,𝔽₅)×#12068, #72242242, …
maximal subgroup of	D₅×#12068, C₅×AGL(1,ℤ/32), #72242242
maximal normal subgroup of	D₅×#12068, C₅×AGL(1,ℤ/32), #72242242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5056);