#27598864

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2810 ⋊ C₂
Sylow	C₅₃ ⋊ #37006

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2810	C₂
prime radical	C₂³×D₅₃	C₂⁴
hypercenter	#2810	D₅₃
hypocenter	C₅₃	#37006
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2810

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 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)(63,69)(67,75)(68,77)(74,76)(78,81,82,90)(79,85,84,89)(80,88,87,93)(83,91,86,92);(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)(54,57,58,56)(55,61,60,59)(64,71)(67,75)(70,72)(74,76)(79,84)(80,87)(85,89)(88,93);(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)(55,60)(59,61)(62,65,66,73)(63,68,69,77)(64,70,71,72)(67,74,75,76)(79,84)(83,86)(85,89)(91,92);(54,56,58,57)(55,59,60,61)(62,64)(63,67)(65,72)(66,71)(68,76)(69,75)(70,73)(74,77)(78,80,82,87)(79,83,84,86)(81,88,90,93)(85,91,89,92);(54,55)(56,59)(57,61)(58,60)(62,63)(64,67)(65,68)(66,69)(70,74)(71,75)(72,76)(73,77)(78,79,82,84)(80,86,87,83)(81,89,90,85)(88,91,93,92)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,i^c=i^52,i^d=i^52,i^e=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, 384340);