C₁₃×#12865 #15699204

order	3328
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12865

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#410
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×#41	C₂³×C₁₃
socle	C₂×C₁₃	#1820
prime radical	C₂×C₁₃×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₄²×C₁₃ ◁ C₄×C₁₃×#41 ◁ G

334 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 106 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)(16,25,28,23)(17,29)(18,22)(19,24)(20,27,35,42)(21,26,38,30)(31,34)(32,40)(33,39)(36,45)(37,41,43,44);(14,16)(15,20,24,35)(17,30)(18,27)(19,28)(21,36,38,34)(22,25,39,23)(26,32)(29,43,40,37)(31,44)(33,42)(41,45);(14,17,18,31,19,32,33,45)(15,21,22,37,24,38,39,43)(16,26,27,41,28,30,42,44)(20,34,25,40,35,36,23,29)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5853);