#27415057

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23843

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2083	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2083	D₅₃
hypocenter	C₅₃	#23843
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2083

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

2^8,53|a^2=eh,b^2=e,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,i^b=i^52,g^c=gh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 200533);