C₅₃×#9427 #27216939

order	13568
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#9427

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

422 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18,23,26,22,27,21,25)(19,28,30,20,24,32,29,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19)(18,20)(21,30)(22,24)(23,29)(25,32)(26,28)(27,31);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(17,20,22,31)(18,24,27,19)(21,32,23,28)(25,29,26,30)

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 2415);