C₅×#12067 #11934787

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12067

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#442
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#322 ◁ G

246 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#12067, AGL(1,𝔽₅)×#12067, #72242236, …
maximal subgroup of	D₅×#12067, C₅×AGL(1,ℤ/32), #72242236, …
maximal normal subgroup of	D₅×#12067, C₅×AGL(1,ℤ/32), #72242236, …
maximal quotient of	C₅×AGL(1,ℤ/32), C₅²×#12067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5055);