C₅₃×#7049 #27214561

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7049

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#95
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₄³	C₂²×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 13568

A minimal faithful permutation representation — 2 generators on 77 points:

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

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, 37);