#27430060

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2079 ⋊ C₂
Sylow	C₅₃ ⋊ #25052

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2079	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20284384	C₂
hypercenter	#2079	D₅₃
hypocenter	C₅₃	#25052
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2079

720 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)(62,65,68,81,67,80,66,79)(63,71,82,92,73,87,72,78)(64,75,83,93,77,91,76,90)(69,84,70,86,85,89,74,88);(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)(55,59)(58,61)(62,64)(63,70)(65,75)(66,76)(67,77)(68,83)(69,72)(71,86)(73,74)(78,84)(79,90)(80,91)(81,93)(82,85)(87,88)(89,92);(54,56,57,60)(55,58,59,61)(62,64,68,83,67,77,66,76)(63,69,82,70,73,85,72,74)(65,75,81,93,80,91,79,90)(71,84,92,86,87,89,78,88);(54,55)(56,58)(57,59)(60,61)(62,63)(64,74)(65,78)(66,82)(67,73)(68,72)(69,76)(70,77)(71,79)(75,89)(80,92)(81,87)(83,85)(84,91)(86,93)(88,90)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 215536);