#27384075

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1801 ⋊ C₂
Sylow	C₅₃ ⋊ #20730

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1801	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1801	D₅₃
hypocenter	C₅₃	#20730
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1801

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

Polycyclic presentation:

2^8,53|a^2=g,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,i^b=i^52,g^c=gh,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, 169551);