C₁₃×#12308 #15698647

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

Aliases

direct product

C₁₃×#12308

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

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

normal subgroups only

This group has 274 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)(14,15,18,21,19,24,35,23)(16,25,27,40,28,43,34,42)(17,29,32,22,33,39,31,41)(20,30,37,26,38,44,36,45);(14,16,19,28)(15,20,24,38)(17,30,33,44)(18,34,35,27)(21,36,23,37)(22,40,41,42)(25,29,43,39)(26,32,45,31);(15,21,24,23)(20,36,38,37)(22,29,41,39)(25,40,43,42)

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, 5296);