#27496202

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #30232

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2099	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#2099	D₅₃
hypocenter	C₅₃	#30232
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2099

558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 93 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)(62,65,66,73)(63,69,67,75)(64,70,71,72)(68,76,74,77)(79,85)(83,91)(84,86)(92,93);(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)(55,59)(58,61)(62,64,66,71)(63,68,67,74)(65,72,73,70)(69,77,75,76)(79,84)(80,87)(81,90)(83,91)(85,86)(88,89);(54,56,57,60)(55,58,59,61)(63,67)(68,74)(69,75)(76,77)(78,80,81,88,82,89,90,87)(79,83,86,92,85,91,84,93);(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)(56,58)(57,59)(60,61)(62,63,66,67)(64,68,71,74)(65,69,73,75)(70,76,72,77)(78,79,81,86,82,85,90,84)(80,83,88,92,89,91,87,93)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 281678);