C₅₃×#12054 #27219566

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#12054

Subgroups

	subgroup	G/H
center	C₂×C₈×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle, prime radical	C₂²×C₅₃	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

206 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 85 points:

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

Conjugacy classes & character table

4664 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 5042);