#27301550

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1383 ⋊ C₂
Sylow	C₅₃ ⋊ #13076

Subgroups

	subgroup	G/H
center	C₂	#20275142
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1383	C₂
socle	C₂×C₅₃	#1893
prime radical	#20284378	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1383	D₅₃
hypocenter	C₅₃	#13076
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

466 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 109 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)(55,63)(56,64)(58,67)(59,79)(60,74)(61,62)(65,72)(66,68)(69,81)(70,84)(71,75)(73,78)(76,85)(77,83)(80,82);(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)(54,56,58,68,59,70,76,63,61,69,78,82,60,71,77,72)(55,62,81,73,80,74,75,83,65,57,64,67,66,79,84,85);(54,55,59,66,61,65,60,80)(56,67,70,83,69,73,71,85)(57,72,74,68,62,63,79,82)(58,75,76,81,78,84,77,64)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=eh,i^b=i^52,d^c=dh,f^c=fh,i^c=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 87026);