C₁₃×#12847 #15699186

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12847

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

434 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 166 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)(17,18)(20,21)(22,23,25,30,26,31,36,43,27,29,35,41,37,42,47,51)(24,28,33,39,34,40,46,50,32,38,44,48,45,49,52,53);(15,18)(19,21)(22,24)(23,28)(25,33)(26,34)(27,32)(29,38)(30,39)(31,40)(35,44)(36,46)(37,45)(41,48)(42,49)(43,50)(47,52)(51,53);(14,16,17,20)(15,19,18,21)(23,29)(24,32)(25,35)(31,42)(34,45)(36,47)(39,48)(50,53)

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, 5835);