#72244338

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×AGL(1,ℤ/8) ⋊ C₂²
Hall	C₃×C₇ ⋊ C₂×#453
Sylow	C₇ ⋊ D₃×#453
Sylow	C₃ ⋊ D₇×#453

Subgroups

	subgroup	G/H
center	C₂	#13060281
commutator, cosocle	C₂×C₃×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₇
Fitting	C₃×C₇×AGL(1,ℤ/8)	C₂²
socle	C₂×C₃×C₇	C₂×#120
hypercenter	AGL(1,ℤ/8)	D₃×D₇
hypocenter	C₃×C₇	C₂×#453
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^7,3,7|f^2=g,b^a=bf,c^a=cg,c^b=ce,d^a=dg,d^b=de,e^a=eg,f^a=fg,f^b=fg,f^c=fg,f^d=fg,h^c=h^2,i^d=i^6

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₇×#453, #72247112, #72269435
maximal normal subgroup of	D₃×D₇×#453, #72269435
maximal quotient of	#72247112

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 7]);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 4, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);