#72245191

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×Q₈ ⋊ C₂×C₃
Hall	#11716073 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ SD₁₆
Sylow	C₇² ⋊ C₃×SD₁₆

Subgroups

	subgroup	G/H
center	C₂	#11924202
commutator, cosocle	C₄×C₇²	C₂²×C₃
Frattini	C₄	C₂×ΓL(1,𝔽₈)⋊D₇
Fitting	C₇²×Q₈	C₂×C₃
socle	C₂×C₇²	C₃×D₄
prime radical	C₄⋊(ΓL(1,𝔽₈)⋊D₇)	C₂
hypercenter	Q₈	ΓL(1,𝔽₈)⋊D₇
hypocenter	C₇²	C₃×SD₁₆
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×ΓL(1,𝔽₈)⋊D₇, #72245202, #72245208, …
maximal normal subgroup of	SD₁₆×ΓL(1,𝔽₈)⋊D₇, #72245202, #72245208, …

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