#27283032

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9671

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1479	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1479	D₅₃
hypocenter	C₅₃	#9671
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1479

434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(54,57,58,59)(55,60,56,61)(62,65,66,76,67,77,79,90)(63,69,71,75,70,83,78,89)(64,72,74,86,73,85,87,91)(68,80,81,84,82,92,88,93);(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)(54,56,59,61,58,55,57,60)(62,64,67,73)(63,68)(65,72,77,85)(66,74,79,87)(69,80)(70,82)(71,81)(75,84)(76,86,90,91)(78,88)(83,92)(89,93);(54,55,59,60,58,56,57,61)(62,63)(64,68,73,82)(65,75)(66,78)(67,70)(69,76)(71,79)(72,84,85,93)(74,88,87,81)(77,89)(80,91,92,86)(83,90)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,c^2=e,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,f^b=fg,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 68508);