C₅₃×#9421 #27216933

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×#9421

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

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

normal subgroups only

This group has 488 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) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,5,7,4,8,3,6)(9,10,14,19,15,18,13,26)(11,16,22,31,21,32,28,30)(12,23,24,17,25,34,27,33)(20,35,38,39,36,40,37,29)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,2,4,8)(3,7,5,6)(9,11,13,22,15,21,14,28)(10,16,19,30,18,32,26,31)(12,20,27,38,25,36,24,37)(17,29,34,40,33,39,23,35);(3,5)(6,7)(9,12)(10,17)(11,20)(13,27)(14,24)(15,25)(16,29)(18,33)(19,34)(21,36)(22,38)(23,26)(28,37)(30,40)(31,35)(32,39)

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, 2409);