#28299367

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7532

Subgroups

	subgroup	G/H
center	C₂	#20289847
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂×C₅₃	#1408
prime radical	C₂²×C₅₃	#353
cosocle	#16811003	C₂²
hypercenter	C₈²	#6157
hypocenter	C₅₃	#7532
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₈²

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1696

Polycyclic presentation:

2^8,53|a^2=d,b^2=h,c^2=fg,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^30,c^b=cfgh,d^b=def,e^b=egh,f^b=fh,i^b=i^52,d^c=dfg,e^d=egh,f^d=fh,i^d=i^52

Conjugacy classes & character table

857 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1084843);