#27538229

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2200 ⋊ C₂
Sylow	C₅₃ ⋊ #33560

Subgroups

	subgroup	G/H
center	C₂	B(4,𝔽₂)×D₅₃
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2200	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	#2200	D₅₃
hypocenter	C₅₃	#33560
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2200

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(1,4,6,16)(2,8,10,22)(3,12,14,15)(5,17,18,31)(7,19,20,21)(9,23,24,32)(11,25,26,29)(13,27,28,30)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84);(4,16)(5,18)(8,22)(12,15)(13,28)(19,21)(23,32)(25,29)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,3,6,14)(2,7,10,20)(4,15,16,12)(5,13,18,28)(8,21,22,19)(9,11,24,26)(17,30,31,27)(23,29,32,25);(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,25)(14,26)(15,29)(16,22)(17,23)(18,24)(19,27)(20,28)(21,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

476 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323705);