#72228678

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#361 ⋊ C₂×C₃
Hall	#11727008 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ AGL(1,ℤ/16)
Sylow	C₇ ⋊ C₃×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂	#89557⋊C₃
commutator	C₇×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#361	C₂×C₃
socle	C₂×C₇	C₃×#442
prime radical	#89919⋊C₃	C₂
cosocle	C₂×C₇×C₈	C₂³×C₃
hypercenter	#361	AGL(1,𝔽₇)
hypocenter	C₇	C₃×AGL(1,ℤ/16)
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

308 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^3,3,2^4,7|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,b^a=bfgh,c^a=cgh,e^a=eh,f^b=fg,f^c=fh,g^b=gh,i^a=i^6,i^b=i^6,i^c=i^6,i^d=i^4

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×AGL(1,ℤ/16)
maximal normal subgroup of	AGL(1,𝔽₇)×AGL(1,ℤ/16)

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