#72244285

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂²
Hall	C₃×C₇ ⋊ C₂×B(4,𝔽₂)
Sylow	C₇ ⋊ D₃×B(4,𝔽₂)
Sylow	C₃ ⋊ D₇×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#13060669
commutator, cosocle	C₂³×C₃×C₇	C₂⁴
Frattini	C₂³	C₂²×D₃×D₇
Fitting	C₃×C₇×H(5,𝔽₂)	C₂²
socle	C₂×C₃×C₇	C₂×#120
hypercenter	H(5,𝔽₂)	D₃×D₇
hypocenter	C₃×C₇	C₂×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

2036 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 2036 conjugacy classes of subgroups — too many to draw. Only its 178 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:

2^7,3,7|a^2=g,b^a=bf,d^b=de,e^a=eg,f^d=fg,h^c=h^2,i^b=i^6,i^c=i^6

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₇×B(4,𝔽₂), #72269444
maximal normal subgroup of	D₃×D₇×B(4,𝔽₂), #72269444

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 7]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 3, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);