#27278922

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8987

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1279	C₂
socle, prime radical	C₂³×C₅₃	#120
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1279	D₅₃
hypocenter	C₅₃	#8987
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1279

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 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)(66,68,69,76)(67,72,71,79)(70,78,77,75)(73,81,80,74);(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)(54,56,57,55)(59,62)(60,65)(63,64)(67,71)(70,77)(72,79)(75,78);(54,55,57,56)(58,59,60,63,61,64,65,62)(66,67)(68,74,76,81)(69,71)(70,73)(72,75,79,78)(77,80)

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 64398);