#72249118

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

Aliases

No aliases for this group.

Splittings

Hall	#13133546 ⋊ C₂
Hall	D₇ ⋊ #6891
Sylow	C₇ ⋊ C₂×#6891

Subgroups

	subgroup	G/H
center, Frattini	C₃²	C₃²×AGL(1,𝔽₇)
commutator, socle, cosocle	C₃²×C₇	C₂×C₃³
Fitting	C₃⁴×C₇	C₂×C₃
prime radical	C₃⁴×D₇	C₃
hypercenter	C₃⁴	AGL(1,𝔽₇)
hypocenter	C₇	C₂×#6891
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃⁴

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 189

Polycyclic presentation:

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

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249123, AGL(1,𝔽₇)×#6891, #72249151
maximal normal subgroup of	#72249123, AGL(1,𝔽₇)×#6891, #72249151

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