#27279591

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9100

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1274	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1274	D₅₃
hypocenter	C₅₃	#9100
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1274

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 65067);