C₅×#12175 #11934895

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12175

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#2190	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

346 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 70 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)(8,18)(9,22)(10,14)(11,16)(12,17)(13,19)(15,25)(20,34)(21,35)(23,28)(24,33)(26,32)(27,31)(29,37)(30,36);(6,8)(7,13)(9,23)(10,20)(11,27)(12,21)(14,28)(15,31)(16,29)(17,30)(18,22)(19,25)(24,37)(26,36)(32,34)(33,35);(7,14,17,16)(9,24)(13,28,30,29)(15,22,32,33)(18,34,35,31)(19,36)(23,37)(25,26)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5163);