C₅×#13658 #11936378

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13658

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2177
commutator	C₁₆	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#2263	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#346 ◁ G

194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 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)(7,13,15,29)(8,18)(9,21)(10,25)(11,19)(14,22,26,30)(16,28,33,32)(17,34,27,31)(20,24)(23,35)(36,37);(6,7)(8,14)(9,22)(10,26)(11,27)(12,15)(13,20)(16,25)(17,19)(18,33)(21,29)(23,28)(24,30)(31,37)(32,36)(34,35);(6,8,19,25,12,10,11,18)(7,14,17,16,15,26,27,33)(9,21,36,35,24,20,23,37)(13,28,31,22,29,32,34,30)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#13658, AGL(1,𝔽₅)×#13658, #72242232, …
maximal subgroup of	D₅×#13658, C₅×AGL(1,ℤ/32), #72242232
maximal normal subgroup of	D₅×#13658, C₅×AGL(1,ℤ/32), #72242232

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6646);