C₅₃×#12512 #27220024

order	13568
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₅₃×#12512

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₅₃×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

268 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,28)(20,29)(21,32)(22,31)(23,30)(24,27)(25,26)(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);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,29,27,23,31,22,30,24,20,28,18,25,32,21,26);(2,7)(6,14)(8,12)(9,15)(17,20,28,27,23,32,21,30,24,19,29,18,25,31,22,26)

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5500);