C₁₃×#12303 #15698642

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12303

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
prime radical	C₁₃×#3204	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×M₁₆ ◁ G

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

normal subgroups only

This group has 322 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)(14,15,18,34,19,24,35,23)(16,25,28,40,29,39,27,21)(17,30,32,22,33,20,31,37)(26,38,44,42,45,41,43,36);(15,20)(17,31)(18,35)(21,36)(22,34)(23,37)(24,30)(25,41)(26,43)(27,28)(32,33)(38,39)(40,42)(44,45);(14,16,19,29)(15,21,24,40)(17,26,33,45)(18,28,35,27)(20,36,30,42)(22,41,37,38)(23,39,34,25)(31,43,32,44)

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, 5291);