#27290774

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#421 ⋊ C₂
Sylow	C₅₃ ⋊ #11003

Subgroups

	subgroup	G/H
center	C₄	#16810531
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#421	C₂
socle	C₂×C₅₃	#1438
prime radical	#20288044	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#421	D₅₃
hypocenter	C₅₃	#11003
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#421

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 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)(54,57)(55,62)(56,66)(58,71)(59,70)(60,72)(61,75)(63,80)(64,79)(65,81)(67,84)(68,83)(69,73)(74,82)(76,85)(77,78);(54,56)(55,61)(57,66)(58,74)(59,67)(60,68)(62,78)(63,76)(64,77)(65,73)(69,85)(70,84)(71,82)(72,83)(75,79)(80,81);(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)(56,65)(57,69,72,80)(58,73)(59,63)(60,64)(61,67)(62,71,79,70)(66,78,83,75)(68,76)(74,77)(81,82,85,84)

Polycyclic presentation:

2^8,53|a^2=g,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,i^a=i^52,c^b=cf,d^b=dh,d^c=dh,e^c=eh,g^c=gh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76250);