#27536753

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2110 ⋊ C₂
Sylow	C₅₃ ⋊ #33329

Subgroups

	subgroup	G/H
center	C₄	C₂×C₄²×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2110	C₂
socle	C₂×C₅₃	C₂²×#97
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2110	D₅₃
hypocenter	C₅₃	#33329
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #2110

716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

1226 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322229);