#27367304

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1690 ⋊ C₂
Sylow	C₅₃ ⋊ #19204

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1690	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×D₅₃	#121
hypercenter	#1690	D₅₃
hypocenter	C₅₃	#19204
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1690

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 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)(65,77)(69,83)(74,88)(78,87)(81,92)(84,90)(85,93)(89,91);(54,55,57,59)(56,61,60,58)(62,65)(63,69)(64,74)(66,78)(67,77)(68,81)(70,84)(71,83)(72,85)(73,87)(75,89)(76,88)(79,90)(80,91)(82,92)(86,93);(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)(62,64,73,80,67,76,66,75)(63,68,79,86,71,82,70,72)(65,74,87,91,77,88,78,89)(69,81,90,93,83,92,84,85);(55,58)(56,60)(59,61)(62,63,67,71)(64,72,76,86)(65,69,77,83)(66,79,73,70)(68,80,82,75)(74,85,88,93)(78,90,87,84)(81,91,92,89)

Polycyclic presentation:

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

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 152780);