C₅₃×#11960 #27219472

order	13568
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#11960

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

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

normal subgroups only

This group has 392 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:

(1,2,3,6,4,5,7,8)(9,10,14,20,15,18,30,36,16,21,28,38,12,27,13,19)(11,22,25,35,23,37,24,32,26,39,29,17,31,33,40,34)(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);(2,5)(3,7)(9,11)(10,17)(12,23)(13,29)(14,24)(15,31)(16,26)(18,32)(19,37)(20,22)(21,35)(25,30)(27,34)(28,40)(33,36)(38,39);(2,5)(6,8)(9,12,16,15)(10,18,21,27)(11,23,26,31)(13,28,30,14)(17,32,35,34)(19,20,36,38)(22,33,39,37)(24,29,40,25)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4948);