#72254209

order	4704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₇² ⋊ C₂×M₁₆
Sylow	C₇² ⋊ C₂×ΓL(1,𝔽₂₅)
Sylow	C₃ ⋊ #13096284

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×D₇²
commutator, socle	C₂×C₃×C₇²	C₂²×C₄
Fitting, cosocle	C₃×C₄×C₇²	C₂³
prime radical	C₂×D₃×C₇⋊D₇	C₄
hypocenter	C₃×C₇²	C₂×M₁₆
LCS	G ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₄

434 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13096284, #72269363, #72269396
maximal normal subgroup of	D₃×#13096284, #72269363, #72269396

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