#27300969

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#504 ⋊ C₂
Sylow	C₅₃ ⋊ #12980

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#13133237	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#12980
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#504

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(58,61)(59,67)(60,74)(62,72)(63,82)(64,78)(65,66)(68,85)(69,80)(70,77)(71,81)(73,89)(75,88)(76,86)(79,84)(83,87);(55,57)(58,60,73,77,62,80,84,67,65,76,83,85,64,75,63,81)(59,66,86,87,68,78,88,82,71,61,74,89,70,72,69,79);(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)(54,55,56,57)(58,59,63,69,64,70,83,74,65,71,84,88,62,68,73,86)(60,72,67,87,75,61,77,79,76,78,81,89,80,66,85,82)

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^52,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

2156 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86445);