#72246136

order	9408
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₇² ⋊ C₂²×C₃
Hall	#72215034 ⋊ C₃
Hall	#3862 ⋊ C₂×#120
Sylow	C₇² ⋊ C₂×C₃×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×#64422⋊C₂
commutator, socle, cosocle	C₂²×C₇²	C₂⁴×C₃
Fitting	C₂⁴×C₇²	C₂²×C₃
hypercenter	C₂⁴	#64422⋊C₂
hypocenter	C₇²	C₂×C₃×#120
LCS	G ▷ C₂²×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂⁴

3022 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 3022 conjugacy classes of subgroups — too many to draw. Only its 277 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) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72246153
maximal normal subgroup of	#72246153

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^6*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.8^6*F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);