C₁₃×#12304 #15698643

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12304

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
prime radical	C₁₃×#3259	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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)(14,15,20,24,19,25,18,35)(16,26,36,43,28,22,29,39)(17,30,31,40,33,21,34,23)(27,37,44,38,32,41,45,42);(15,21)(17,31)(18,20)(22,37)(23,35)(24,40)(25,30)(26,41)(27,44)(29,36)(32,45)(33,34)(38,43)(39,42);(14,16,19,28)(15,22)(17,32,33,27)(18,29,20,36)(21,37)(23,38)(24,39)(25,26)(30,41)(31,45,34,44)(35,43)(40,42)

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, 5292);