C₅×#12303 #11935023

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12303

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#3204	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

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

normal subgroups only

This group has 322 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)(6,7,10,26,11,16,27,15)(8,17,20,32,21,31,19,13)(9,22,24,14,25,12,23,29)(18,30,36,34,37,33,35,28);(7,12)(9,23)(10,27)(13,28)(14,26)(15,29)(16,22)(17,33)(18,35)(19,20)(24,25)(30,31)(32,34)(36,37);(6,8,11,21)(7,13,16,32)(9,18,25,37)(10,20,27,19)(12,28,22,34)(14,33,29,30)(15,31,26,17)(23,35,24,36)

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, 5291);