#72245348

order	7056
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×C₈ ⋊ C₂×C₃²
Hall	#11716062 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ SD₁₆
Sylow	C₇² ⋊ C₃²×SD₁₆

Subgroups

	subgroup	G/H
center	C₂	#72222014
commutator, cosocle	C₄×C₇²	C₂²×C₃²
Frattini	C₄	C₂×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
Fitting	C₇²×C₈	C₂×C₃²
socle	C₂×C₇²	C₃²×D₄
prime radical	#72222014	C₂
hypercenter	C₈	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×SD₁₆
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Conjugacy classes & character table

127 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72245379, #72245389
maximal normal subgroup of	SD₁₆×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72245379, #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, 3, 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, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.7^5*F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);