#27416162

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2028 ⋊ C₂
Sylow	C₅₃ ⋊ #23953

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2028	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288436	C₂
hypercenter	#2028	D₅₃
hypocenter	C₅₃	#23953
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2028

1166 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)(62,66,68,67)(63,71,72,80)(64,74,75,81)(65,77,79,78)(69,83,85,84)(70,76,87,88)(73,90,91,92)(82,93,86,89);(62,65,68,79)(63,70,72,87)(64,73,75,91)(66,78,67,77)(69,82,85,86)(71,88,80,76)(74,92,81,90)(83,89,84,93);(54,56,57,61)(55,58,59,60)(62,64)(63,69)(65,73)(66,74)(67,81)(68,75)(70,86)(71,83)(72,85)(76,89)(77,90)(78,92)(79,91)(80,84)(82,87)(88,93);(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)(54,55)(56,60)(57,59)(58,61)(62,63)(64,69)(65,76)(66,71)(67,80)(68,72)(70,78)(73,89)(74,83)(75,85)(77,87)(79,88)(81,84)(82,90)(86,92)(91,93)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201638);