#72249208

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₇×AΣL(1,𝔽₂₇) ⋊ C₂×C₃
Hall	ΓL(1,𝔽₈)×AΣL(1,𝔽₂₇) ⋊ C₂
Sylow	C₇ ⋊ C₃×#4183

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×AΣL(1,𝔽₂₇)	C₂×C₃
Frattini	C₃²	C₃⋊(C₃⋊AGL(1,𝔽₇))
socle	C₃×C₇	C₃×#272
LCS	G ▷ C₇×AΣL(1,𝔽₂₇)
UCS	C₁

300 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 63

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72249208, #72254880, AGL(1,𝔽₇)×#4183
maximal normal subgroup of	C₃×#72249208, #72254880, AGL(1,𝔽₇)×#4183
maximal quotient of	C₃×#72249208

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