C₅₃×#12301 #27219813

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12301

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
prime radical	C₅₃×#3199	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×M₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 342 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:

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

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5289);