#72222210

order	2646
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃×C₇² ⋊ C₂×C₃²
Hall	C₃×ΓL(1,𝔽₈)² ⋊ C₂
Sylow	C₇² ⋊ C₃²×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇²	C₂×C₃²
LCS	G ▷ C₃×C₇²
UCS	C₁

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^3,7^2|d^a=d^2,e^a=e^6,e^b=f,e^c=e^2,f^a=f^6,f^b=e^5f^3,f^c=f^2

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222215, D₃×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72224671, …
maximal normal subgroup of	D₃×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), C₃×#72222210, AGL(1,𝔽₇)×C₃⋊AGL(1,𝔽₇), …
maximal quotient of	#72222215, #72224671, #72230226, …

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