#27271588

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1321 ⋊ C₂
Sylow	C₅₃ ⋊ #7398

Subgroups

	subgroup	G/H
center	C₂	#20272713
commutator	C₂²×C₈×C₅₃	C₂×C₄
Frattini	#408	C₂×D₅₃
Fitting	C₅₃×#1321	C₂
socle	C₂×C₅₃	#1343
prime radical	#20275824	C₂
cosocle	C₅₃×#408	C₂²
hypercenter	#1321	D₅₃
hypocenter	C₅₃	#7398
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

258 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 2 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)(55,60)(56,66)(57,70)(58,67)(61,64)(62,79)(63,75)(65,77)(69,83)(71,72)(73,82)(74,80)(78,81)(84,85);(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)(54,55,57,62)(56,65,68,79)(58,64,72,80)(59,63,71,81)(60,73,74,83)(61,76,78,69)(66,75,82,85)(67,77,70,84)

Polycyclic presentation:

2^8,53|a^2=d,c^2=fh,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,c^b=cf,d^b=def,f^b=fh,i^b=i^52,d^c=dg,e^d=eh

Conjugacy classes & character table

701 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57064);