#27295847

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1373 ⋊ C₂
Sylow	C₅₃ ⋊ #12046

Subgroups

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1373	C₂
socle	C₂×C₅₃	C₈×D₈
prime radical	#16811055	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1373	D₅₃
hypocenter	C₅₃	#12046
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(54,57,58,71,60,73,76,85,61,74,66,80,59,72,75,84)(55,62,64,78,65,79,68,81,56,67,69,82,70,83,63,77);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56)(55,61)(57,67)(58,69)(59,65)(60,70)(62,74)(63,76)(64,66)(68,75)(71,82)(72,79)(73,83)(77,85)(78,80)(81,84);(54,55,58,64,60,65,76,68,61,56,66,69,59,70,75,63)(57,62,71,78,73,79,85,81,74,67,80,82,72,83,84,77)

Polycyclic presentation:

2^8,53|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,d^b=df,f^b=fh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81323);