#72222285

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃×C₇ ⋊ C₂×C₃×D₃
derived	C₃×C₇×A₄ ⋊ C₂²×C₃
Sylow	C₇ ⋊ C₃×D₃×S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₇×A₄	C₂²×C₃
Fitting, socle	C₂²×C₃×C₇	C₂×C₃×D₃
LCS	G ▷ C₃×C₇×A₄
UCS	C₁

326 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 33 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:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×S₄×AGL(1,𝔽₇), #72244658
maximal normal subgroup of	D₃×S₄×AGL(1,𝔽₇)
maximal quotient of	#72244658

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