C₅₃×#13692 #27221204

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13692

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6680);