#27538197

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2200 ⋊ C₂
Sylow	C₅₃ ⋊ #33556

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	#33556
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2200

1134 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:

(1,5,7,4)(2,10,12,9)(3,14,16,18)(6,19,20,17)(8,21,22,24)(11,25,26,23)(13,27,28,32)(15,29,31,30)(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);(1,4,7,5)(2,9,12,10)(3,14,16,18)(6,19,20,17)(8,21,22,24)(11,23,26,25)(13,27,28,32)(15,30,31,29)(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);(1,3,7,16)(2,8,12,22)(4,14,5,18)(6,15,20,31)(9,21,10,24)(11,13,26,28)(17,29,19,30)(23,27,25,32);(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,27)(16,28)(17,23)(18,32)(19,25)(20,26)(21,29)(22,31)(24,30)

Polycyclic presentation:

2^8,53|b^2=h,c^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,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, 323673);