#27286312

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1995 ⋊ C₂
Sylow	C₅₃ ⋊ #10215

Subgroups

	subgroup	G/H
center	C₂²	#16810532
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1995	C₂
socle	C₂²×C₅₃	#392
prime radical	#16811173	C₂²
hypercenter	#1995	D₅₃
hypocenter	C₅₃	#10215
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1995

464 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 3392

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)(54,56)(55,60)(57,66)(58,63)(59,65)(61,69)(62,68)(64,67)(70,73,75,71)(72,79,81,85)(74,84,76,78)(77,80,83,82);(54,55)(56,64)(57,61)(58,67)(59,62)(60,66)(63,68)(65,69)(70,72,75,81)(71,77,73,83)(74,82,76,80)(78,85,84,79);(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)(56,63)(57,58)(60,64)(61,67)(65,66)(68,69)(70,71)(72,79)(73,75)(74,84)(76,78)(77,80)(81,85)(82,83)

Polycyclic presentation:

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

Conjugacy classes & character table

860 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 71788);