#72246766

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₇×C₂²⋊C₄ ⋊ C₂
Hall	C₃²×C₇ ⋊ #120
Sylow	C₃² ⋊ #6445
Sylow	C₇ ⋊ #64091

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₂₁
commutator, socle, cosocle	C₂²×C₃²×C₇	C₂³
Fitting	C₃²×C₇×C₂²⋊C₄	C₂
hypercenter	C₂²⋊C₄	C₃⋊D₂₁
hypocenter	C₃²×C₇	#120
LCS	G ▷ C₂²×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

780 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 504

Polycyclic presentation:

2^5,3^2,7|b^a=bd,c^a=ce,f^a=f^2,f^b=f^2,g^a=g^2,g^b=g^2,h^a=h^6,h^b=h^6

Conjugacy classes & character table

324 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#6445, #120×C₃⋊D₂₁, D₇×#64091, …
maximal normal subgroup of	C₃⋊D₃×#6445, #120×C₃⋊D₂₁, D₇×#64091, …

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