C₅₃×#7121 #27214633

order	13568
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7121

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#102
commutator	C₂×C₈	C₂×C₈×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

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

normal subgroups only

This group has 222 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) = 13568

A minimal faithful permutation representation — 2 generators on 85 points:

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

Conjugacy classes & character table

3074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 109);