#72213636

order	10584
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃²×C₇² ⋊ C₂³×C₃
Hall	C₃²×#3861 ⋊ C₂³
Sylow	C₇² ⋊ C₂×C₃×D₃²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₇²	C₂³×C₃
LCS	G ▷ C₃²×C₇²
UCS	C₁

876 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃²×#64423⋊C₂, C₂×#72213636, C₃×#72213636, …
maximal normal subgroup of	D₃²×#64423⋊C₂, C₂×#72213636, C₃×#72213636
maximal quotient of	#72222196, #72222230, #72222236, …
holomorph of	D₂₁

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