C₅₃×#19094 #27226606

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#19094

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

910 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 3392

A minimal faithful permutation representation — 4 generators on 85 points:

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

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 12082);