#72230002

order	6804
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₇×C₉² ⋊ C₂²×C₃
Hall	#13133594 ⋊ C₂²
Sylow	C₇ ⋊ C₂×#92314

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉²	C₂²×C₃
Frattini	C₃²	C₃⋊D₃×AGL(1,𝔽₇)
socle	C₃²×C₇	C₂×C₃×C₃⋊D₃
LCS	G ▷ C₇×C₉²
UCS	C₁

508 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 567

Polycyclic presentation:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#92314, #72230192
maximal normal subgroup of	AGL(1,𝔽₇)×#92314, #72230192

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