#72267527

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

No aliases for this group.

Splittings

derived

C₂²⋊(C₂²⋊SL(2,𝔽₃)) ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×#87588
commutator, hypocenter	C₂²⋊(C₂²⋊SL(2,𝔽₃))	C₂×C₄
Fitting	C₂⁵×#41	D₃
socle	C₂⁶	C₂×S₄
prime radical	C₂×#11710913	C₂
cosocle	C₂×C₂²⋊(C₂²⋊SL(2,𝔽₃))	C₂²
hypercenter	C₂×C₄	#87588
LCS	G ▷ C₂²⋊(C₂²⋊SL(2,𝔽₃))
UCS	C₁ ◁ C₂² ◁ C₂×C₄

82700 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 82700 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 96

Polycyclic presentation:

2^3,3,2^7|a^2=c,b^2=k,g^2=k,h^2=k,b^a=bk,d^a=d^2,e^a=f,e^d=ef,f^a=e,f^d=e,g^a=fhk,g^d=gh,h^a=egk,h^d=g,h^g=hk,i^a=fij,i^d=j,j^a=efj,j^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267529
maximal normal subgroup of	#72267529

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