C₅₃×#13463 #27220975

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13463

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×D₄×C₅₃ ◁ G

484 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 77 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,22)(21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,3,5,4)(2,6,7,8)(10,14)(13,20)(15,21)(17,23);(1,4,5,3)(2,6,7,8)(9,11,12,18)(10,15,14,21)(13,19,20,24)(16,17,22,23)

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6451);