C₅₃×#10428 #27217940

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#10428

Subgroups

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

386 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

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, 3416);