#30743408

order	17152
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₆₇ ⋊ #7425

Subgroups

	subgroup	G/H
center	C₂	#22558219
commutator	C₂×C₁₆×C₆₇	C₂×C₄
Frattini	C₂×M₃₂	C₂×D₆₇
Fitting	C₆₇×#1400	C₂
socle	C₂×C₆₇	#1347
prime radical	C₂³×C₆₇	SD₃₂
cosocle	C₂×M₃₂×C₆₇	C₂²
hypercenter	#1400	D₆₇
hypocenter	C₆₇	#7425
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #1400

136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 4288

Polycyclic presentation:

2^8,67|a^2=d,b^2=h,c^2=fgh,d^2=h,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,i^a=i^66,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,i^b=i^66,d^c=dh

Conjugacy classes & character table

1483 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17152, 57145);