C₅₃×#7034 #27214546

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#7034

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#95
commutator	C₂³	C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₄²	C₂²×C₅₃
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

406 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 2 generators on 77 points:

(1,2,3,5,4,6,7,8)(9,10)(11,15,16,13)(12,14)(18,21)(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);(9,11)(10,13)(12,16)(14,15)(17,18,19,22,20,21,23,24)

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, 22);