#27538676

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2223 ⋊ C₂
Sylow	C₅₃ ⋊ #33618

Subgroups

	subgroup	G/H
center	C₂	#466×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2223	C₂
socle	C₂×C₅₃	C₂×#466
hypercenter	#2223	D₅₃
hypocenter	C₅₃	#33618
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2223

656 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)(55,64)(61,74)(63,79)(65,66)(73,84)(75,76)(78,85)(80,81);(54,56)(55,63)(57,68)(58,72)(59,67)(60,69)(61,73)(62,77)(64,79)(65,81)(66,80)(70,83)(71,82)(74,84)(75,85)(76,78);(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)(54,55,62,76,59,66,71,74,60,64,70,75,58,65,57,61)(56,63,77,78,67,80,82,84,69,79,83,85,72,81,68,73);(55,61)(56,67)(57,62)(58,59)(63,78)(64,74)(65,76)(66,75)(68,82)(69,72)(70,71)(73,80)(79,85)(81,84)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324152);