C₅₃×#10309 #27217821

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#10309

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

618 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 81
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 81 points:

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

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 3297);