#72209928

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₃×C₄

solvable

Aliases

No aliases for this group.

Splittings

Fitting

#585 ⋊ #157

Subgroups

	subgroup	G/H
center, hypercenter	C₄	#10596770
commutator, hypocenter	Ω⁺(4,𝔽₃)	C₂×C₄
Frattini, socle	C₂	C₂×#10596770
Fitting	#585	#157
prime radical	#11923520	C₂
cosocle	SO⁺(4,𝔽₃)	C₂²
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₄

267 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^2,2^5|a^2=j,b^2=c,f^2=j,g^2=j,h^2=j,i^2=j,d^b=de,d^c=d^2,e^b=de^2,e^c=e^2,f^b=i,f^c=fj,f^d=g,f^e=g,g^b=h,g^c=fg,g^d=fgj,g^e=fgj,g^f=gj,h^b=fg,h^c=hij,h^d=hi,i^b=fj,i^c=ij,i^d=h,i^h=ij

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209929
maximal quotient of	#72227892

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.4*F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.6*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.8);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);