C₅₃×#9456 #27216968

order	13568
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#9456

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

432 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2,5,10,6,9,4,8)(3,7,11,14,12,15,13,16)(17,18,22,25,21,26,20,24)(19,23,29,31,28,32,27,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,11,6,12,5,13)(2,7,10,16,9,15,8,14)(17,18,21,26)(19,23,28,32)(20,25,22,24)(27,31,29,30);(1,4,6,5)(2,8,9,10)(3,11,12,13)(7,14,15,16)(17,19,21,28)(18,23,26,32)(20,29,22,27)(24,31,25,30)

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 2444);