C₅×#12306 #11935026

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12306

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#469	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

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

normal subgroups only

This group has 274 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)(6,7,12,16,11,17,10,27)(8,18,28,35,20,14,21,31)(9,22,23,32,25,13,26,15)(19,29,36,30,24,33,37,34);(7,13)(9,23)(10,12)(14,29)(15,27)(16,32)(17,22)(18,33)(19,36)(21,28)(24,37)(25,26)(30,35)(31,34);(6,8)(7,14,17,18)(9,24)(10,21)(11,20)(12,28)(13,29,22,33)(15,30,32,34)(16,31,27,35)(19,25)(23,37)(26,36)

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, 5294);