C₅₃×#7339 #27214851

order	13568
nilpotency class	5
isoclinism	isoclinic class of #7339
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#7339

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1316
commutator	C₂²×C₄	C₂×C₈×C₅₃
Frattini, cosocle	#381	C₂²×C₅₃
prime radical	C₂²×#41×C₅₃	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂³×C₅₃ ◁ C₂⁴×C₅₃ ◁ C₅₃×#381 ◁ G

356 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 85 points:

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

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327);