C₁₃×#12897 #15699236

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12897

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	#432
prime radical	C₂³×C₁₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

278 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 142 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)(16,18)(17,19)(20,21)(22,23,27,33,26,32,42,49,28,34,44,30,43,31,25,40)(24,35,38,50,37,29,36,45,39,46,51,53,52,47,41,48);(14,16,17,20)(15,18,19,21)(22,24,28,39)(23,29)(25,41,42,36)(26,37,43,52)(27,38,44,51)(30,48)(31,35)(32,46)(33,45)(34,47)(40,50)(49,53);(15,19)(18,21)(22,25,26,27,28,42,43,44)(23,30,31,49,34,33,32,40)(24,36,37,51,39,41,52,38)(29,45,35,50,47,48,46,53)

Conjugacy classes & character table

832 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5885);