#27537036

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#360 ⋊ C₂
Sylow	C₅₃ ⋊ #33388

Subgroups

	subgroup	G/H
center	C₂	C₂×#102×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂×C₅₃	C₂²×#102
prime radical	#20289484	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	#33388
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

1128 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)(56,64)(61,76)(65,78)(66,79)(73,84)(77,82)(80,83)(81,85);(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)(54,56)(55,61)(57,65)(58,66)(59,64)(60,73)(62,76)(63,77)(67,80)(68,81)(69,78)(70,79)(71,82)(72,83)(74,84)(75,85);(54,55)(56,61)(57,68)(58,71)(59,62)(60,72)(63,70)(64,76)(65,81)(66,82)(67,74)(69,75)(73,83)(77,79)(78,85)(80,84);(55,60,62,74)(57,67)(58,70)(61,73,76,84)(63,68,71,75)(65,80)(66,79)(69,72)(77,81,82,85)(78,83)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322512);