#72245205

order	4704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×D₈ ⋊ C₂×C₃
Hall	#13096551 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ AGL(1,ℤ/8)
Sylow	C₇² ⋊ C₃×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	D₄×ΓL(1,𝔽₈)⋊D₇
commutator, cosocle	C₄×C₇²	C₂³×C₃
Frattini	C₄	C₂²×ΓL(1,𝔽₈)⋊D₇
Fitting	C₇²×D₈	C₂×C₃
socle	C₂×C₇²	C₂×C₃×D₄
hypercenter	D₈	ΓL(1,𝔽₈)⋊D₇
hypocenter	C₇²	C₃×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

340 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 68 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:

2^3,3,2^2,7^2|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^b=g^6,g^c=g^6,g^d=h,h^b=h^6,h^c=h^6,h^d=g^5h^3

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊D₇, #72245389
maximal normal subgroup of	AGL(1,ℤ/8)×ΓL(1,𝔽₈)⋊D₇, #72245389

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