C₅₃×#9004 #27216516

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

Aliases

direct product

C₅₃×#9004

Subgroups

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

458 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 89
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 89 points:

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

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, 1992);