#72247124

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ #41
Sylow	C₇ ⋊ C₃×#3796

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×(C₃⋊AGL(1,𝔽₇))⋊D₃
commutator, socle, cosocle	C₂×C₃²×C₇	C₂³×C₃
Fitting	C₃²×C₄×C₇	C₂²×C₃
hypercenter	C₄	(C₃⋊AGL(1,𝔽₇))⋊D₃
hypocenter	C₃²×C₇	C₃×#41
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₄

420 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 79 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:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247133, AGL(1,𝔽₇)×#3796, #72247162, …
maximal normal subgroup of	#72247133, AGL(1,𝔽₇)×#3796, #72247162, …

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