C₅×#11522 #11934242

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11522

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	#1464
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×M₁₆ ◁ G

284 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,9,15,10,16,22,32,11,14,21,30,23,13,27,29)(8,17,19,31,20,28,34,37,18,12,24,26,33,25,35,36);(6,8)(7,12)(9,19)(10,20)(11,18)(13,28)(14,17)(15,31)(16,25)(21,24)(22,34)(23,33)(26,30)(27,35)(29,36)(32,37);(7,13,14,16)(8,18)(9,21)(12,25,17,28)(15,32,30,29)(20,33)(22,27)(26,37,31,36)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4510);