C₁₃×#12313 #15698652

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12310
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12313

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
prime radical	C₁₃×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₄×C₈×C₁₃ ◁ G

356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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)(14,15,20,16)(17,27,30,43)(18,32,37,33)(19,26,35,23)(21,38,24,29)(22,41,25,36)(28,42,31,44)(34,39,45,40);(14,16,20,15)(17,24,30,21)(18,33,37,32)(19,23,35,26)(22,36,25,41)(27,34,43,45)(28,39,31,40)(29,44,38,42);(14,17)(15,21)(16,24)(18,34)(19,28)(20,30)(22,42)(23,39)(25,44)(26,40)(27,32)(29,36)(31,35)(33,43)(37,45)(38,41)

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, 5301);