#27276347

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1488 ⋊ C₂
Sylow	C₅₃ ⋊ #8476

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1488	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288637	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1488	D₅₃
hypocenter	C₅₃	#8476
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1488

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 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)(54,56,58,64)(55,60,59,62)(57,66,65,63)(61,69,67,68)(70,73,75,86)(71,77,79,84)(72,81,83,98)(74,89,90,85)(76,91,92,100)(78,95,87,93)(80,96,94,101)(82,99,88,97);(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)(55,59)(57,65)(60,62)(63,66)(70,72)(71,76)(73,81)(74,88)(75,83)(77,91)(78,94)(79,92)(80,87)(82,90)(84,100)(85,99)(86,98)(89,97)(93,96)(95,101);(54,55,57,61,58,59,65,67)(56,62,63,69,64,60,66,68)(70,71)(72,80)(73,84,86,77)(74,87)(75,79)(76,82)(78,90)(81,96,98,101)(83,94)(85,93,89,95)(88,92)(91,97,100,99)

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61823);