C₁₃×#7413 #15693752

order	3328
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#7413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#354
commutator	C₂×C₄²	C₂×C₄×C₁₃
Frattini, cosocle	#391	C₂²×C₁₃
prime radical	C₂³×C₁₃	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#391 ◁ G

178 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 53 points:

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 401);