C₅₃×#12117 #27219629

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#12117

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅₃
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×M₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 424 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) = 85
transitive degree	μ_t(G) = 1696

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

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

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, 5105);