#27433217

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1816 ⋊ C₂
Sylow	C₅₃ ⋊ #25265

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1816	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811130	C₂²
hypercenter	#1816	D₅₃
hypocenter	C₅₃	#25265
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1816

728 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)(55,58)(59,61)(62,66,68,67)(63,71,73,72)(64,75,77,76)(65,80,81,82)(69,84,86,85)(70,78,88,89)(74,91,79,92)(83,90,87,93);(54,56,57,60)(55,59,58,61)(62,65)(63,70)(64,74)(66,80)(67,82)(68,81)(69,83)(71,78)(72,89)(73,88)(75,91)(76,92)(77,79)(84,90)(85,93)(86,87);(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)(55,58)(59,61)(62,64,67,76,68,77,66,75)(63,69,72,85,73,86,71,84)(65,79,80,92,81,74,82,91)(70,87,78,93,88,83,89,90);(54,55)(56,59)(57,58)(60,61)(62,63,67,72,68,73,66,71)(64,69,76,85,77,86,75,84)(65,78,80,88,81,89,82,70)(74,90,91,87,79,93,92,83)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 218693);