#27436721

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1882 ⋊ C₂
Sylow	C₅₃ ⋊ #25529

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1882	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811130	C₂²
hypercenter	#1882	D₅₃
hypocenter	C₅₃	#25529
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1882

1014 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)(55,59)(58,60)(65,78)(70,88)(73,90)(76,89)(77,91)(79,82)(83,92)(87,93);(62,65,66,79,68,78,67,82)(63,70,71,76,72,88,80,89)(64,73,74,91,75,90,81,77)(69,83,84,87,86,92,85,93);(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)(54,56,57,61)(55,58,59,60)(62,64,68,75)(63,69,72,86)(65,77,78,91)(66,74,67,81)(70,87,88,93)(71,84,80,85)(73,82,90,79)(76,92,89,83);(54,55)(56,60)(57,59)(58,61)(62,63)(64,69)(65,76)(66,80)(67,71)(68,72)(70,79)(73,87)(74,85)(75,86)(77,92)(78,89)(81,84)(82,88)(83,91)(90,93)

Polycyclic presentation:

2^8,53|b^2=eh,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,i^b=i^52,d^c=dh,g^c=gh,i^d=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 222197);