C₅₃×#23357 #27230869

order	13568
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅₃×#23357

Subgroups

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

912 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 306 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24,20,22,19,23)(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);(3,8)(5,7)(10,13)(11,15)(18,22)(23,24);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(17,19,20,21)(18,23,22,24);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,19,20,21)(18,23,22,24)

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 16345);