#72245653

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×Q₈ ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×SD₁₆
Sylow	C₇ ⋊ C₃×D₃×SD₁₆

Subgroups

	subgroup	G/H
center	C₂	#66694⋊C₃
commutator, cosocle	C₃×C₄×C₇	C₂³×C₃
Frattini	C₄	C₂×D₃×AGL(1,𝔽₇)
Fitting	C₃×C₇×Q₈	C₂²×C₃
socle	C₂×C₃×C₇	C₂×C₃×D₄
prime radical	#66681⋊C₃	C₂
hypercenter	Q₈	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×SD₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

292 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 168

Polycyclic presentation:

2^3,3,2^2,3,7|a^2=e,e^2=f,b^a=bef,e^b=ef,g^c=g^2,h^a=h^6,h^b=h^6,h^c=h^6,h^d=h^4

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245660, D₃×SD₁₆×AGL(1,𝔽₇), #72245671, …
maximal normal subgroup of	#72245660, D₃×SD₁₆×AGL(1,𝔽₇), #72245671
maximal quotient of	#72254200

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 3, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);