C₅₃×#12764 #27220276

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#12764

Subgroups

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

768 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2,3,4)(5,6)(7,15,18,30)(8,12)(9,22,24,13)(10,14)(11,17)(16,33,19,32)(20,31,36,28)(21,29)(23,26)(25,34)(27,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7,10,18)(6,11,14,26)(8,19,21,16)(9,23,24,17)(12,27,29,25)(13,30,22,15)(20,34,36,35)(28,33,31,32);(5,8,10,21)(6,12,14,29)(7,16,18,19)(9,20,24,36)(11,25,26,27)(13,31,22,28)(15,32,30,33)(17,34,23,35)

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, 5752);