C₁₃×#13607 #15699946

order	3328
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13607

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#2127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	#124	C₂³×C₁₃
prime radical	C₁₃×#535	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#535 ◁ G

332 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 45
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,24,26,39)(17,28,29,44)(18,31,33,22)(19,23)(20,25,35,42)(21,30)(27,37,32,40)(34,41,36,43)(38,45);(14,16,19,27)(15,20,23,34)(17,25,29,41)(18,32,33,26)(21,37,38,24)(22,35,31,36)(28,40,44,39)(30,42,45,43);(17,29)(20,34)(21,38)(25,41)(28,44)(30,45)(35,36)(42,43)

Conjugacy classes & character table

325 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6595);