#72216992

order	3072
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

derived

#11710730 ⋊ C₄

Subgroups

	subgroup	G/H
center, socle	C₂	#72216975
commutator, hypocenter	#11710730	C₄
Frattini	C₂⁵	#1020
Fitting	#6499867	D₃
prime radical, cosocle	#72209717	C₂
hypercenter	C₂²	#11714267
LCS	G ▷ #11710730
UCS	C₁ ◁ C₂ ◁ C₂²

2731 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 2731 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^8|a^2=b,d^2=k,e^2=k,c^a=c^2,d^a=dgjk,d^b=dik,d^c=e,e^a=defj,e^b=ehk,e^c=dek,e^d=ek,f^a=fgh,f^c=g,f^d=fj,f^e=fj,g^a=gi,g^c=fg,g^e=gj,h^a=hi,h^c=hi,h^d=hk,i^c=h,i^e=ik,j^a=jk

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218442, #72218483, C₂×#72216992
maximal normal subgroup of	#72218418, #72218442, #72218483, …
maximal quotient of	#72241394, C₂×#72216992

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