C₁₃×#13606 #15699945

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×#13606

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#2127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	#124	C₂³×C₁₃
prime radical	C₁₃×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#536 ◁ G

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

normal subgroups only

This group has 320 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)(14,15,17,23)(16,21,26,35)(18,22,28,37)(19,24,29,38)(20,30,33,31)(25,36,39,45)(27,34,41,44)(32,42,43,40);(15,20)(17,28)(18,31)(21,34)(22,24)(26,41)(27,40)(29,30)(32,36)(33,37)(35,45)(39,42);(14,16,19,25)(15,21,24,36)(17,26,29,39)(18,27,31,40)(20,32,22,34)(23,35,38,45)(28,41,30,42)(33,43,37,44)

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, 6594);