#72269288

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

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₇² ⋊ SD₁₆
Sylow	C₇² ⋊ #214
Sylow	C₃ ⋊ #11716036

Subgroups

	subgroup	G/H
center	C₂	#11924253
commutator, Fitting, cosocle	C₃×C₄×C₇²	C₂²
Frattini, hypercenter	C₄	D₂₁⋊D₇
socle	C₂×C₃×C₇²	D₄
prime radical	C₃×C₄⋊(C₇⋊D₇)	C₂
hypocenter	C₃×C₇²	SD₁₆
LCS	G ▷ C₃×C₄×C₇² ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269290, D₃×#11716036, #72269302, …
maximal normal subgroup of	#72269290, D₃×#11716036, #72269302, …

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