#27365301

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1660 ⋊ C₂
Sylow	C₅₃ ⋊ #18991

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1660	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288861	C₂
hypercenter	#1660	D₅₃
hypocenter	C₅₃	#18991
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1660

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150777);