C₅₃×#8133 #27215645

order	13568
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₅₃ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₅₃×#8133

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator	C₂²	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ G

446 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 13568

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

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

Conjugacy classes & character table

5936 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1121);