#27537985

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2190 ⋊ C₂
Sylow	C₅₃ ⋊ #33530

Subgroups

	subgroup	G/H
center	C₂	#447×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2190	C₂
socle	C₂×C₅₃	C₂×#447
hypercenter	#2190	D₅₃
hypocenter	C₅₃	#33530
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #2190

1132 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:

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

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,f^b=fh,g^b=gh,e^c=eh,i^d=i^52

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323461);