#27365359

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1664 ⋊ C₂
Sylow	C₅₃ ⋊ #18999

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1664	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×D₄×D₅₃	D₄
hypercenter	#1664	D₅₃
hypocenter	C₅₃	#18999
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1664

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 101 points:

(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)(71,78)(76,91)(77,92)(79,86)(80,94)(90,101)(93,95)(96,97);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56)(55,59)(57,67)(58,63)(60,69)(61,68)(62,65)(64,66)(70,73,75,85)(71,77,78,92)(72,81,83,99)(74,88,89,84)(76,90,91,101)(79,95,86,93)(80,96,94,97)(82,100,87,98);(57,66)(60,65)(62,69)(64,67)(70,72)(71,76)(73,81)(74,87)(75,83)(77,90)(78,91)(79,94)(80,86)(82,89)(84,100)(85,99)(88,98)(92,101)(93,96)(95,97);(54,55,58,61)(56,62,63,69)(57,65,66,60)(59,67,68,64)(70,71,75,78)(72,80,83,94)(73,77,85,92)(74,86,89,79)(76,87,91,82)(81,97,99,96)(84,95,88,93)(90,100,101,98)

Polycyclic presentation:

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

Conjugacy classes & character table

554 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150835);