#27538005

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3414 ⋊ C₂
Sylow	C₅₃ ⋊ #33533

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810755
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3414	C₂
socle	C₂×C₅₃	C₂×#447
hypercenter	#3414	D₅₃
hypocenter	C₅₃	#33533
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

1102 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)(54,56,59,64)(55,61,62,74)(57,65,69,77)(58,66,70,78)(60,73,72,83)(63,75,71,81)(67,79,82,85)(68,80,76,84);(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)(56,64)(57,69)(60,72)(61,74)(66,78)(67,82)(68,76)(75,81);(54,55)(56,61)(57,68)(58,71)(59,62)(60,67)(63,70)(64,74)(65,80)(66,81)(69,76)(72,82)(73,79)(75,78)(77,84)(83,85);(55,60)(57,67)(58,70)(61,73)(62,72)(63,76)(65,79)(66,78)(68,71)(69,82)(74,83)(75,84)(77,85)(80,81)

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,d^c=dh,e^c=eh,i^c=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323481);