#27540674

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2189 ⋊ C₂
Sylow	C₅₃ ⋊ #33839

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2189	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2189	D₅₃
hypocenter	C₅₃	#33839
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2189

712 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)(59,65)(60,71)(61,75)(62,76)(64,81)(67,68)(69,80)(70,88)(72,73)(74,83)(77,84)(78,79)(82,85)(87,89);(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)(60,70)(64,80)(69,81)(71,88)(72,79)(73,78)(82,87)(85,89);(58,60)(59,64)(61,71)(62,78)(63,70)(65,82)(66,85)(67,87)(68,80)(69,74)(72,75)(73,76)(77,88)(79,84)(81,83)(86,89);(54,55)(56,57)(58,59,63,68)(60,69,70,81)(61,74,77,83)(62,65,76,67)(64,79,80,72)(66,84,86,75)(71,87,88,82)(73,89,78,85)

Polycyclic presentation:

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

Conjugacy classes & character table

797 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326150);