C₅×#11836 #11934556

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11836

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1514
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#469	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×M₁₆ ◁ G

286 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,18,20,35)(9,23,25,30)(10,16)(11,14)(12,17)(13,19)(15,26,32,27)(21,31,22,34)(24,33)(28,37)(29,36);(6,8)(7,13)(9,24)(10,22)(11,21)(12,20)(14,28)(15,31)(16,33)(17,29)(18,23)(19,27)(25,37)(26,36)(30,35)(32,34);(7,14,17,16)(9,25)(13,28,29,33)(15,23,32,30)(18,34,35,31)(19,36)(24,37)(26,27)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4824);