#72222126

order	7056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃×#3861 ⋊ C₂×D₄
Sylow	C₇² ⋊ C₂×C₃×#67

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×#64423⋊C₂
commutator, Fitting, socle, cosocle	C₂×C₃×C₇²	C₂³×C₃
hypocenter	C₃×C₇²	C₂×C₃×D₄
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂

703 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72210376, #72222241, C₂×#72222126
maximal normal subgroup of	D₃×#72210376, C₂×#72222126
maximal quotient of	#72222241, C₂×#72222126

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