C₁₃×#12327 #15698666

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12327

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
prime radical	C₁₃×#3277	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ C₁₃×#436 ◁ G

368 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 53
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,19)(17,20)(18,21)(22,23,28,32)(24,33,36,51)(25,30,27,42)(26,38,41,49)(29,44,46,35)(31,37,50,40)(34,52,43,48)(39,45,53,47);(15,19)(16,17)(20,21)(22,24)(23,29)(25,34)(26,39)(27,43)(28,36)(30,45)(31,48)(32,46)(33,38)(35,40)(37,44)(41,53)(42,47)(49,51)(50,52);(15,20)(16,17)(19,21)(22,25,28,27)(23,30,32,42)(24,34,36,43)(26,40,41,37)(29,45,46,47)(31,49,50,38)(33,48,51,52)(35,53,44,39)

Conjugacy classes & character table

403 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5315);