#27289679

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×H(3,ℤ/4) ⋊ C₂
Sylow	C₅₃ ⋊ #10792

Subgroups

	subgroup	G/H
center	C₄	C₄×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×H(3,ℤ/4)	C₂
socle	C₂×C₅₃	C₂²⋊#95
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	#10792
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×H(3,ℤ/4)

1204 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)(55,61)(56,64)(58,72)(59,68)(60,74)(62,77)(63,79)(65,82)(66,83)(67,70)(69,73)(71,81)(75,84)(76,85)(78,80);(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)(54,56)(55,60,62,76)(57,64)(58,71)(59,65)(61,74,77,85)(63,78,70,75)(66,73)(67,84,79,80)(68,82)(69,83)(72,81);(54,55)(56,60,65,76)(57,67)(58,70)(59,62)(61,69)(63,73)(64,80,82,84)(66,78,71,75)(68,79)(72,77)(74,83,85,81)

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 75155);