C₅₃×#23355 #27230867

order	13568
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₅₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅₃×#23355

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

984 conjugacy classes of subgroups; only the 386 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 386 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) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 77 points:

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

Conjugacy classes & character table

4028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 16343);