#30607444

order	15616
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₆₁×#362 ⋊ C₄
Sylow	C₆₁ ⋊ #12954

Subgroups

	subgroup	G/H
center	C₂	#21461784
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#362	C₄
socle	C₂×C₆₁	#1864
prime radical	C₂²×D₆₁	C₂×SD₁₆
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#362	#6923
hypocenter	C₆₁	#12954
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

380 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 3904

Polycyclic presentation:

2^8,61|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,i^a=i^50,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^11,d^c=dgh,e^c=egh,f^c=fh,g^c=gh,i^f=i^60

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1096999);