#72252802

order	10368
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ #1844

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂²	#72213247
commutator	#10620150	C₂²×C₄
Fitting, socle	C₂²×C₃⁴	#127
prime radical	C₂²×C₃⋊(C₃⋊(C₃⋊D₃))	C₂×D₄
cosocle	C₂×#10620150	C₂³
hypocenter	C₃⁴	#1844
LCS	G ▷ #10620150 ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂²

1912 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1912 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 192

Polycyclic presentation:

2^7,3^4|a^2=d,b^2=e,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,h^b=i,h^c=i,h^e=h^2,h^f=h^2,i^b=h^2,i^c=h^2,i^e=i^2,i^f=i^2,j^a=j^2,j^b=h^2ij,j^c=k,j^e=hj,j^f=j^2,k^b=h^2i^2k,k^c=j^2,k^e=ik,k^f=k^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252832
maximal normal subgroup of	#72252832

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.8^2);;
gap> SetConjugate(coll, 9, 3, F.8^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 2, F.8^2*F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 10, 5, F.8*F.10);;
gap> SetConjugate(coll, 10, 6, F.10^2);;
gap> SetConjugate(coll, 11, 2, F.8^2*F.9^2*F.11);;
gap> SetConjugate(coll, 11, 3, F.10^2);;
gap> SetConjugate(coll, 11, 5, F.9*F.11);;
gap> SetConjugate(coll, 11, 6, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);