#72222014

order	3528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₇² ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)² ⋊ D₄
Hall	C₄⋊(C₇⋊D₇) ⋊ C₃²
Sylow	C₇² ⋊ C₃²×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₇²	C₂²×C₃²
Fitting	C₄×C₇²	C₂×C₃²
hypercenter	C₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×D₄
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄

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

normal subgroups only

This group has 224 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

Polycyclic presentation:

2^2,3^2,2,7^2|a^2=e,b^a=be,f^b=f^6,f^c=f^5g^3,f^d=f^4,g^b=g^6,g^c=f,g^d=g^4

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), C₂×#72222014, #72245321, …
maximal normal subgroup of	D₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), C₂×#72222014, #72245321, …
maximal quotient of	C₂×#72222014, #72245348, #72245365, …

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