#27358107

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1600 ⋊ C₂
Sylow	C₅₃ ⋊ #18306

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1600	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#1600	D₅₃
hypocenter	C₅₃	#18306
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1600

1040 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)(54,57,58,64)(55,60,61,67)(56,62,63,68)(59,66,65,69)(71,76)(75,84)(77,81)(78,85);(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)(55,59)(57,62)(58,63)(60,66)(61,65)(64,68)(67,69)(73,83)(77,81)(78,85)(79,82);(54,55)(56,59)(57,60)(58,61)(62,66)(63,65)(64,67)(68,69)(70,72)(71,75)(73,82)(74,80)(76,84)(77,85)(78,81)(79,83);(57,64)(59,65)(60,67)(62,68)(70,71)(72,78)(73,81)(74,76)(75,79)(77,83)(80,85)(82,84)

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 143583);