#72249155

order	10206
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#6910 ⋊ C₂×C₃
Hall	#19102203 ⋊ C₂
Hall	D₇ ⋊ #10624749
Sylow	C₇ ⋊ C₂×#10624749

Subgroups

	subgroup	G/H
center	C₃	#72249122
commutator, cosocle	C₃³×C₇	C₂×C₃³
Frattini	C₃³	C₃²×AGL(1,𝔽₇)
Fitting	C₇×#6910	C₂×C₃
socle	C₃×C₇	C₂×C₃²⋊H(3,𝔽₃)
hypercenter	#6910	AGL(1,𝔽₇)
hypocenter	C₇	C₂×#10624749
LCS	G ▷ C₃³×C₇ ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6910

976 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 107 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:

2,3^6,7|b^3=g^2,c^3=g^2,d^3=g^2,d^b=de,d^c=def^2,e^b=eg,e^c=eg^2,e^d=eg^2,f^b=fg^2,f^c=fg^2,f^d=fg,h^a=h^6,h^b=h^2

Conjugacy classes & character table

149 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#10624749
maximal normal subgroup of	AGL(1,𝔽₇)×#10624749

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