C₁₃×#12309 #15698648

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12309

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
prime radical	C₁₃×#469	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

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

normal subgroups only

This group has 280 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)(14,15,20,24,19,25,18,22)(16,26,28,43,30,44,29,40)(17,31,34,41,33,39,35,23)(21,36,37,27,38,32,42,45);(14,16)(15,21)(17,32)(18,28)(19,30)(20,29)(22,37)(23,40)(24,42)(25,38)(26,39)(27,34)(31,44)(33,36)(35,45)(41,43);(15,22,25,24)(17,33)(21,37,38,42)(23,39,41,31)(26,43,44,40)(27,45)(32,36)(34,35)

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, 5297);