C₅×#12412 #11935132

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12412

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#446 ◁ G

280 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,12,16)(8,17,9,20)(10,15,11,14)(13,19,21,18)(22,23,28,32)(24,33,36,51)(25,30,27,42)(26,38,41,49)(29,44,46,35)(31,37,50,40)(34,52,43,48)(39,45,53,47);(6,8,12,9)(7,13,16,21)(10,19,11,18)(14,20,15,17)(22,24,28,36)(23,29,32,46)(25,34,27,43)(26,39,41,53)(30,45,42,47)(31,48,50,52)(33,49,51,38)(35,37,44,40);(6,9,12,8)(7,13,16,21)(10,18,11,19)(14,20,15,17)(22,25,28,27)(23,30,32,42)(24,34,36,43)(26,40,41,37)(29,45,46,47)(31,49,50,38)(33,48,51,52)(35,53,44,39)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5400);