#27537951

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2184 ⋊ C₂
Sylow	C₅₃ ⋊ #33526

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2184	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2184	D₅₃
hypocenter	C₅₃	#33526
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2184

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,60,63,69)(59,65,68,79)(61,70,74,82)(62,71,73,83)(64,76,78,87)(66,80,77,88)(67,81,75,85)(72,84,86,89);(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)(55,57)(60,69)(65,79)(70,82)(71,83)(76,87)(80,88)(81,85)(84,89);(59,64)(61,72)(62,73)(65,76)(66,67)(68,78)(70,84)(71,83)(74,86)(75,77)(79,87)(80,81)(82,89)(85,88);(54,55)(56,57)(58,59)(60,65)(61,64)(62,75)(63,68)(66,72)(67,73)(69,79)(70,76)(71,85)(74,78)(77,86)(80,84)(81,83)(82,87)(88,89)

Polycyclic presentation:

2^8,53|d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh,i^d=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323427);