#30772947

order	17152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₆₇ ⋊ #13020

Subgroups

	subgroup	G/H
center	C₂²	#16863529
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1979	C₂
socle	C₂²×C₆₇	#447
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1979	D₆₇
hypocenter	C₆₇	#13020
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #1979

460 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 8576

Polycyclic presentation:

2^8,67|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^66,d^b=dg,e^b=eg,f^b=fg,g^b=gh,i^b=i^66,d^c=dg,e^c=egh,f^c=fg,g^c=gh,i^c=i^66

Conjugacy classes & character table

1102 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 86684);