#72244251

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄²×C₇ ⋊ C₂×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ #127
Sylow	C₇ ⋊ C₃×#874

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₃×C₇	C₂³×C₃
Fitting	C₃×C₄²×C₇	C₂×C₃
hypercenter	C₄²	C₃⋊AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₃×#127
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄²

462 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 336

Polycyclic presentation:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#127×C₃⋊AGL(1,𝔽₇), D₃×#10620710, AGL(1,𝔽₇)×#874, …
maximal normal subgroup of	#127×C₃⋊AGL(1,𝔽₇), D₃×#10620710, AGL(1,𝔽₇)×#874, …
maximal quotient of	#72246448

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, 1, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 7, 2, 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);