C₁₃×#12884 #15699223

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#12884

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

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

normal subgroups only

This group has 302 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)(22,23,27,33,28,32,44,49,29,34,43,31,26,45,25,41)(24,35,38,50,39,47,36,48,40,51,52,53,37,30,42,46);(15,18)(19,21)(22,24,26,37,29,40,28,39)(23,30,32,35,34,47,45,51)(25,42,43,52,44,36,27,38)(31,48,41,53,33,46,49,50);(14,16,17,20)(15,19,18,21)(22,25,26,43,29,44,28,27)(23,31,32,41,34,33,45,49)(24,36,37,38,40,42,39,52)(30,46,35,50,47,48,51,53)

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, 5872);