C₅₃×#8242 #27215754

order	13568
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#8242

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

384 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 6784

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

(1,2,4,7)(3,9,10,6)(5,13,12,8)(11,14,16,15)(17,18,21,25,22,26,20,23)(19,24,28,31,29,32,27,30)(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);(1,3,5,11,4,10,12,16)(2,6,8,14,7,9,13,15)(18,23,26,25)(24,30,32,31);(17,19,21,28,22,29,20,27)(18,24,25,31,26,32,23,30)

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, 1230);