C₁₃×#12326 #15698665

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

Aliases

direct product

C₁₃×#12326

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃
prime radical	C₁₃×#3273	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄²×C₁₃ ◁ C₁₃×#436 ◁ G

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

normal subgroups only

This group has 390 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) = 832

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)(22,23,27,31)(24,32,34,50)(25,36,38,52)(26,41,39,30)(28,43,44,33)(29,46,48,40)(35,51,42,47)(37,49,53,45);(15,19)(16,17)(20,21)(22,24)(23,28)(25,37)(26,42)(27,34)(29,47)(30,49)(31,44)(32,36)(33,40)(35,39)(38,53)(41,45)(43,46)(48,51)(50,52);(15,20)(16,17)(19,21)(25,38)(29,48)(32,50)(33,43)(36,52)(37,53)(40,46)(47,51)

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, 5314);