#72245545

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²×C₇ ⋊ C₂²×C₃
Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂×D₄
Sylow	C₇ ⋊ C₃×D₄×C₃⋊D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊D₃×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₃²×C₇	C₂³×C₃
Fitting	C₂²×C₃²×C₇	C₂²×C₃
hypercenter	C₂²	C₃⋊D₃×AGL(1,𝔽₇)
hypocenter	C₃²×C₇	C₂×C₃×D₄
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₂²

776 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 252

Polycyclic presentation:

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

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×C₃⋊D₃×AGL(1,𝔽₇), C₂×#72245545, D₃×#66694⋊C₃, …
maximal normal subgroup of	D₄×C₃⋊D₃×AGL(1,𝔽₇), C₂×#72245545, D₃×#66694⋊C₃, …
maximal quotient of	C₂×#72245545, #72246464, #72247206, …

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