#72228331

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

#6630948 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#11713628
commutator, cosocle	C₄×C₄²⋊C₃	C₂⁴
Frattini	C₂²×C₄	C₂³×S₄
Fitting	#6630948	D₃
socle	C₂³	C₂×D₄×S₄
hypercenter	M₁₆	#857⋊C₂
hypocenter	C₄²⋊C₃	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄²⋊C₃ ▷ C₂×C₄²⋊C₃ ▷ C₄²⋊C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

7136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 4 generators on 20 points:

(2,6)(5,7)(9,14,18)(10,19,11)(12,20,17)(13,16,15);(2,5)(3,8)(6,7)(10,14)(12,16)(18,19);(1,2,3,7,4,6,8,5)(10,13,14,17)(15,18,20,19);(10,13)(14,17)(15,18)(19,20)

Polycyclic presentation:

2^6,3,2^4|b^2=e,e^2=f,h^2=j,i^2=k,b^a=bj,c^a=cj,c^b=ce,d^b=df,e^c=ef,g^a=g^2,g^b=g^2jk,g^c=gj,g^e=gjk,h^a=hj,h^b=hj,h^c=hj,h^g=ij,i^a=hi,i^b=hi,i^c=ik,i^g=hi,j^g=k,k^a=jk,k^b=jk,k^g=jk

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258767
maximal normal subgroup of	#72258767

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