#27544245

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2357 ⋊ C₂
Sylow	C₅₃ ⋊ #34189

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2357	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#11719947	C₂²
hypercenter	#2357	D₅₃
hypocenter	C₅₃	#34189
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2357

2342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 81 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)(66,69,70,78)(67,73,72,77)(68,76,75,81)(71,79,74,80);(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)(68,75)(71,74)(76,81)(79,80);(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)(58,61,62,60)(59,63,64,65)(67,72)(71,74)(73,77)(79,80);(58,60,62,61)(59,63,64,65)(66,68,70,75)(67,71,72,74)(69,76,78,81)(73,79,77,80);(54,55,56,57)(58,59)(60,63)(61,65)(62,64)(66,67)(68,74)(69,77)(70,72)(71,75)(73,78)(76,79)(80,81)

Polycyclic presentation:

2^8,53|a^2=h,b^2=g,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,i^c=i^52,i^d=i^52,i^e=i^52

Conjugacy classes & character table

1388 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 329721);