#72246553

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₇² ⋊ C₂²
Hall	C₃×C₇² ⋊ C₂×D₄
Sylow	C₇² ⋊ D₃×D₄
Sylow	C₃ ⋊ D₇×D₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁⋊D₇
commutator, socle, cosocle	C₂×C₃×C₇²	C₂³
Fitting	C₃×C₄×C₇²	C₂²
hypercenter	C₄	D₂₁⋊D₇
hypocenter	C₃×C₇²	C₂×D₄
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂ ◁ C₄

264 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 51 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:

2^4,3,7^2|b^a=bd,c^b=cd,e^a=e^2,f^a=f^6,f^b=f^6,f^c=f^6,g^a=f^2g,g^b=g^6,g^c=g^6

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×D₂₁⋊D₇, D₃×D₇×D₂₈, D₇×#66681, …
maximal normal subgroup of	D₄×D₂₁⋊D₇, D₃×D₇×D₂₈, D₇×#66681, …

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