#27540840

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2179 ⋊ C₂
Sylow	C₅₃ ⋊ #33854

Subgroups

	subgroup	G/H
center	C₂	D₈×D₂₁₂
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2179	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2179	D₅₃
hypocenter	C₅₃	#33854
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2179

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326316);