#27350023

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1467 ⋊ C₂
Sylow	C₅₃ ⋊ #17371

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1467	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1467	D₅₃
hypocenter	C₅₃	#17371
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

1098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(62,66,67,79)(63,70,71,85)(64,74,75,91)(65,76,77,78)(68,81,72,87)(69,82,83,84)(73,88,89,90)(80,92,86,93);(55,58)(59,61)(62,65,67,77)(63,69,71,83)(64,73,75,89)(66,78,79,76)(68,80,72,86)(70,84,85,82)(74,90,91,88)(81,93,87,92);(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)(62,64)(63,68)(65,73)(66,74)(67,75)(69,80)(70,81)(71,72)(76,88)(77,89)(78,90)(79,91)(82,92)(83,86)(84,93)(85,87);(54,55,56,59,57,58,60,61)(62,63)(64,72,75,68)(65,69)(66,70)(67,71)(73,86,89,80)(74,87,91,81)(76,82)(77,83)(78,84)(79,85)(88,93,90,92)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=g,f^2=h,b^a=be,c^a=ch,e^a=eg,f^b=fg,i^b=i^52,d^c=dg,i^d=i^52

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135499);