#27361025

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1638 ⋊ C₂
Sylow	C₅₃ ⋊ #18605

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1638	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811170	C₂²
hypercenter	#1638	D₅₃
hypocenter	C₅₃	#18605
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1638

866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 146501);