#27350226

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #17404

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1472	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1472	D₅₃
hypocenter	C₅₃	#17404
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1472

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 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)(54,58,59,57)(55,62,63,61)(56,66,67,65)(60,69,64,68)(70,73,75,74)(71,77,78,82)(72,80,79,81)(76,84,83,85);(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)(54,57,59,58)(55,61,63,62)(56,65,67,66)(60,68,64,69)(72,79)(76,83)(80,81)(84,85);(54,56,59,67)(55,60,63,64)(57,65,58,66)(61,68,62,69)(70,72)(71,76)(73,80)(74,81)(75,79)(77,84)(78,83)(82,85);(54,55)(56,64,67,60)(57,61)(58,62)(59,63)(65,69,66,68)(70,71,74,77,75,78,73,82)(72,76,81,84,79,83,80,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=g,d^2=gh,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,f^b=fg,i^c=i^52,i^d=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135702);