#27465623

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2077 ⋊ C₂
Sylow	C₅₃ ⋊ #27637

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2077	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂×#13133191	C₂²
hypercenter	#2077	D₅₃
hypocenter	C₅₃	#27637
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #2077

730 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)(62,64,65,69)(63,67,66,68)(72,79)(75,83)(80,85)(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)(56,60)(58,61)(71,76)(73,81)(75,82)(77,78)(80,85)(83,84);(54,56)(55,58)(57,60)(59,61)(63,66)(67,68)(70,72)(71,75)(73,80)(74,79)(76,82)(77,84)(78,83)(81,85);(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)(54,55,57,59)(56,58,60,61)(62,63)(64,68)(65,66)(67,69)(70,71,74,78)(72,75,79,83)(73,76,81,77)(80,82,85,84)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 251099);