#72244191

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₄×C₇ ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ #120
Sylow	C₇ ⋊ C₃×#882

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₃×C₇	C₂³×C₃
Fitting	C₂×C₃×C₄×C₇	C₂²×C₃
hypercenter	C₂×C₄	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₃×#120
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

554 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 91 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:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244193, D₃×#6445⋊C₃, AGL(1,𝔽₇)×#882, …
maximal normal subgroup of	#72244193, D₃×#6445⋊C₃, AGL(1,𝔽₇)×#882
maximal quotient of	#72254363, #72254364, #72254442, …

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