C₅₃×#12414 #27219926

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#12414

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₂×D₈×C₅₃	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₅₃×#446 ◁ G

286 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Conjugacy classes & character table

1643 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5402);