C₅₃×#12307 #27219819

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12307

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
prime radical	C₅₃×#3253	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×M₁₆×C₅₃ ◁ G

318 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 848

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

(1,2,5,10,6,11,4,8)(3,12,14,9,15,7,13,16)(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);(2,7)(3,13)(4,5)(8,16)(9,10)(11,12)(14,15);(2,8,11,10)(7,16,12,9)

Conjugacy classes & character table

1325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5295);