#72227543

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#2867 ⋊ ΓL(1,𝔽₈)
derived	C₄³⋊C₇ ⋊ C₂×C₃
Hall	#2867⋊C₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄³⋊C₇	C₂×C₃
Frattini, socle	C₂³	C₂×AΓL(1,𝔽₈)
Fitting	#2867	ΓL(1,𝔽₈)
LCS	G ▷ C₄³⋊C₇
UCS	C₁

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 28 points:

(1,4)(2,14,18,5,13,21)(6,15,8)(7,10,11)(12,19,28)(16,24,25)(17,22,26,23,20,27);(1,2,6,12,11,17,19)(3,8,14,20,18,25,26)(4,5,10,16,15,23,24)(7,13,22,21,28,27,9)

Polycyclic presentation:

2,3,7,2^6|d^2=g,e^2=h,f^2=i,b^a=bgh,c^a=ch,c^b=c^2,d^a=dg,d^b=ehi,d^c=de,e^a=eh,e^b=dfh,e^c=ef,f^a=fi,f^b=deghi,f^c=d,g^b=h,g^c=gh,h^b=gi,h^c=hi,i^b=gh,i^c=g

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72227543
maximal normal subgroup of	C₂×#72227543
maximal quotient of	C₂×#72227543

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