#72260015

order	9408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	#72259476 ⋊ C₃
Hall	#3861 ⋊ #442
Sylow	C₇² ⋊ C₃×#442

Subgroups

	subgroup	G/H
center	C₂²	D₄×#64424
commutator	C₄×C₇²	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×#64424
Fitting	C₂×C₇²×D₈	C₂×C₃
socle	C₂²×C₇²	C₂×C₃×D₄
prime radical	C₂×D₈×#3861	C₂
cosocle	C₂×C₄×C₇²	C₂³×C₃
hypercenter	C₂×D₈	#64424
hypocenter	C₇²	C₃×#442
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

1320 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 268 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) = 1568

Polycyclic presentation:

2^3,3,2^3,7^2|a^2=e,b^2=e,c^2=f,f^2=g,c^a=cf,c^b=cg,f^a=fg,h^a=h^6,h^b=h^6,h^d=h^4,i^a=i^6,i^b=i^6,i^d=i^4

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

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