C₅₃×#13052 #27220564

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13052

Subgroups

	subgroup	G/H
center	C₄×C₅₃	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×#41	C₂³×C₅₃
socle	C₂×C₅₃	#1889
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×#41×C₅₃ ◁ G

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6040);