#30606643

order	15616
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₆₁×#361 ⋊ C₄
Sylow	C₆₁ ⋊ #12651

Subgroups

	subgroup	G/H
center	C₂	#21461590
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#361	C₄
socle	C₂×C₆₁	#1762
prime radical	C₂×#14519796	C₂²
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#361	#6923
hypocenter	C₆₁	#12651
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

412 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 1952

Polycyclic presentation:

2^8,61|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,i^a=i^11,c^b=cg,e^b=eh,i^b=i^50,d^c=dh,f^c=fh,i^c=i^11,i^e=i^60,i^f=i^60

Conjugacy classes & character table

286 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1096198);