#72249134

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#731 ⋊ C₂×C₃
Hall	#13133632 ⋊ C₂
Hall	D₇ ⋊ #6910
Sylow	C₇ ⋊ C₂×#6910

Subgroups

	subgroup	G/H
center	C₃	H(3,𝔽₃)×AGL(1,𝔽₇)
commutator, cosocle	C₃²×C₇	C₂×C₃³
Frattini	C₃²	C₃²×AGL(1,𝔽₇)
Fitting	C₇×#731	C₂×C₃
socle	C₃×C₇	C₂×C₃×H(3,𝔽₃)
prime radical	H(3,𝔽₃)×AGL(1,𝔽₇)	C₃
hypercenter	#731	AGL(1,𝔽₇)
hypocenter	C₇	C₂×#6910
LCS	G ▷ C₃²×C₇ ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

272 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 189

Polycyclic presentation:

2,3^5,7|d^3=f^2,d^b=df^2,d^c=de,e^c=ef,e^d=ef,g^a=g^6,g^b=g^2

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#6910, #72269769
maximal normal subgroup of	AGL(1,𝔽₇)×#6910, #72269769

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