#27283022

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9669

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator	C₂×C₄×C₅₃	C₂²×C₈
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	#9669
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1479

446 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)(62,65,76,90,67,78,66,77)(63,69,79,89,70,83,71,75)(64,72,85,91,73,86,74,87)(68,80,88,93,82,92,81,84);(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)(54,56,57,61,58,55,59,60)(62,64,67,73)(63,68)(65,72,78,86)(66,74,76,85)(69,80)(70,82)(71,81)(75,84)(77,87,90,91)(79,88)(83,92)(89,93);(54,55,57,60,58,56,59,61)(62,63)(64,68,73,82)(65,75)(66,79)(67,70)(69,77)(71,76)(72,84,86,93)(74,88,85,81)(78,89)(80,91,92,87)(83,90)

Polycyclic presentation:

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

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 68498);