C₅₃×#13682 #27221194

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13682

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₅₃
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄⋊Q₈×C₅₃ ◁ G

384 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

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, 6670);