#72246780

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#6628896 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1020⋊C₂
commutator, cosocle	C₂×#997	C₂⁴
Frattini	C₂⁴	C₂³×S₄
Fitting	#6628896	D₃
hypercenter	C₂×D₄	#1020⋊C₂
hypocenter	#997	H(5,𝔽₂)
LCS	G ▷ C₂×#997 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×D₄

9165 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 9165 conjugacy classes of subgroups — too many to draw. Only its 111 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:

2^5,3,2^5|d^2=e,g^2=k,h^2=k,b^a=bk,c^b=ce,d^a=de,d^b=de,d^c=de,f^d=f^2,g^c=gik,g^d=gh,g^f=h,h^c=hijk,h^d=hk,h^f=ghk,h^g=hk,i^d=j,i^f=ij,i^h=ik,j^d=i,j^f=i,j^g=jk,j^h=jk

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247039, #72252563, #72267056, …
maximal normal subgroup of	#72247039, #72252563, #72267056, …
maximal quotient of	C₂×#72246780

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