#27350094

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1473 ⋊ C₂
Sylow	C₅₃ ⋊ #17383

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1473	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂×#13133190	C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1473	D₅₃
hypocenter	C₅₃	#17383
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1473

1020 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)(54,57,58,65)(55,60,61,67)(56,63,64,69)(59,66,62,68);(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)(72,79)(75,78)(80,81)(83,85);(54,56)(55,59)(57,63)(58,64)(60,66)(61,62)(65,69)(67,68)(70,72)(71,75)(73,81)(74,79)(76,78)(77,85)(80,82)(83,84);(54,55)(56,62,64,59)(57,60)(58,61)(63,68,69,66)(65,67)(70,71,73,77,74,76,82,84)(72,78,80,85,79,75,81,83)

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135570);