#72277232

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#518 ⋊ C₂×C₃
Hall	#11729405 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #1904
Sylow	C₇ ⋊ C₃×#1904

Subgroups

	subgroup	G/H
center	C₂	#89985⋊C₃
commutator	C₂×C₄×C₇	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#518	C₂×C₃
socle	C₂×C₇	C₃×#386
prime radical	C₂×#6495⋊C₃	C₂
cosocle	C₂²×C₄×C₇	C₂³×C₃
hypercenter	#518	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#1904
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #518

692 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

121 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#1904
maximal normal subgroup of	AGL(1,𝔽₇)×#1904

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