#72267941

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×GL(2,ℤ/4)
commutator	C₂×#41⋊C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂³×2₋¹⁺⁴	D₃
prime radical	C₂³×2₋¹⁺⁴⋊C₃	C₂
cosocle	C₂²×#41⋊C₃	C₂³
hypercenter	C₂³×D₄	S₄
hypocenter	SL(2,𝔽₃)	#379
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×D₄

6943 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 6943 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 128

Polycyclic presentation:

2^6,3,2^3|c^2=d,d^2=j,e^2=j,h^2=j,i^2=j,c^a=ce,c^b=cf,d^a=dj,e^a=ej,g^c=g^2,h^c=hi,h^g=i,i^c=ij,i^g=hij,i^h=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267959
maximal quotient of	#72267959

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