C₅₃×#11475 #27218987

order	13568
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅₃×#11475

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#99
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#127×C₅₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

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

normal subgroups only

This group has 616 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) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 69 points:

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

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, 4463);