C₅₃×#13057 #27220569

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

Aliases

direct product

C₅₃×#13057

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₅₃
prime radical	C₂×C₅₃×#454	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×Q₈×C₅₃ ◁ G

700 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 73 points:

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

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6045);