C₅₃×#7056 #27214568

order	13568
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#7056

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#99
commutator	C₂×C₄	C₂×C₁₆×C₅₃
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#323
prime radical	C₂²×D₄×C₅₃	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄⋊C₈×C₅₃ ◁ G

248 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 3392

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

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

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, 44);