#72209730

order	2058
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₇

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow	H(3,𝔽₇) ⋊ C₂×C₃
Hall	#11759980 ⋊ C₂
Hall	(C₇⋊D₇)⋊C₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₇)	C₂×C₃
Frattini, socle	C₇	#64423
LCS	G ▷ H(3,𝔽₇)
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,7^3|c^a=ce^2,c^b=c^2e^5,d^a=d^6,d^b=d^2e,d^c=de,e^a=e^6,e^b=e^4

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72209732

GAP

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

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