#72342888

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72312304 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×C₄⋊(#120⋊C₃)
commutator	#585	C₂²×C₃×C₄
Frattini	C₂³×C₄	C₂³×A₄
Fitting	#72312304	C₃
prime radical	C₂²×#3027⋊C₃	C₂
cosocle	C₂×#585	C₂³×C₃
hypercenter	C₂×M₁₆	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₂×C₃×C₂²⋊C₄
LCS	G ▷ #585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

12282 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 12282 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64

Polycyclic presentation:

2^3,3,2^7|c^2=e,e^2=k,f^2=k,c^a=ck,c^b=cf,e^b=ek,f^b=fk,g^b=gi,g^d=h,h^b=hj,h^d=gh,i^d=j,i^h=ik,j^d=ijk,j^g=jk

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342916, #72343876
maximal normal subgroup of	#72342916, #72343876

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