#27276265

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1516 ⋊ C₂
Sylow	C₅₃ ⋊ #8462

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1516	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288357	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1516	D₅₃
hypocenter	C₅₃	#8462
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1516

1014 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61741);