C₅₃×#8779 #27216291

order	13568
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#8779

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#381
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

466 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 3392

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

(1,2,4,6,5,7,3,8)(9,10)(11,17,19,14)(12,15,20,22)(13,16)(18,23)(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);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(1,2,5,7)(3,8,4,6)(9,12,13,20)(10,15)(11,18,19,24)(14,21)(16,22)(17,23)

Conjugacy classes & character table

2279 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1767);