#27276635

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1512 ⋊ C₂
Sylow	C₅₃ ⋊ #8528

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1512	C₂
socle	C₂²×C₅₃	#379
prime radical	#20289067	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1512	D₅₃
hypocenter	C₅₃	#8528
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1512

728 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)(54,55)(56,59)(57,61)(58,60)(62,65)(63,69)(64,73)(66,80)(67,78)(68,82)(70,86)(71,76)(72,88)(74,91)(75,90)(77,81)(79,89)(83,92)(84,87)(85,93);(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)(54,55)(56,61)(57,59)(58,60)(62,64)(63,68)(65,73)(66,79)(67,75)(69,82)(70,85)(71,83)(72,87)(74,81)(76,92)(77,91)(78,90)(80,89)(84,88)(86,93);(55,59)(56,57)(60,61)(62,63)(64,72,75,85)(65,76,78,69)(66,70)(67,71)(68,74,83,79)(73,88)(77,86,80,84)(81,87)(82,89)(90,93)(91,92)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62111);