C₅₃×#7309 #27214821

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#7309

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#342
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₂²×#41	C₂²×C₅₃
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×#41×C₅₃ ◁ G

470 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 3392

A minimal faithful permutation representation — 2 generators on 93 points:

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

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, 297);