#27543016

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2335 ⋊ C₂
Sylow	C₅₃ ⋊ #34081

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2335	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂×#13133189	C₂²
hypercenter	#2335	D₅₃
hypocenter	C₅₃	#34081
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2335

2202 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)(66,69,70,77)(67,73,72,79)(68,76,75,81)(71,78,74,80);(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)(58,60,61,64)(59,63,62,65)(67,72)(71,74)(73,79)(78,80);(59,62)(63,65)(68,75)(71,74)(76,81)(78,80);(54,56,57,55)(66,68)(67,71)(69,76)(70,75)(72,74)(73,78)(77,81)(79,80);(54,55,57,56)(58,59)(60,63)(61,62)(64,65)(66,67)(68,74)(69,73)(70,72)(71,75)(76,80)(77,79)(78,81)

Polycyclic presentation:

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

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 328492);