#72230000

order	5292
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₇²×C₉ ⋊ C₂²×C₃
Hall	C₉×#3861 ⋊ C₂²
Sylow	C₇² ⋊ C₂×C₃×D₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇²×C₉	C₂²×C₃
Frattini	C₃	#13183634
socle	C₃×C₇²	C₂×C₃×D₃
LCS	G ▷ C₇²×C₉
UCS	C₁

196 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 126

Polycyclic presentation:

2^2,3^2,7^2,3|d^3=g,d^a=d^2g^2,e^a=e^6,e^b=e^6,e^c=e^4,f^a=e^2f,f^b=f^6,f^c=f^4,g^a=g^2

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230009, D₉×#64423⋊C₂
maximal normal subgroup of	#72230009, D₉×#64423⋊C₂

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