#72277459

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#518 ⋊ C₂×C₃
Hall	#11742936 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3490
Sylow	C₇ ⋊ C₃×#3490

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁵×C₃
Fitting	C₇×#518	C₂×C₃
hypercenter	#518	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3490
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #518

2656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 23 points:

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

Polycyclic presentation:

2^5,3,2^2,7|b^2=g,c^a=cg,d^c=dh,e^b=eh,i^a=i^6,i^f=i^2

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3490, #72277632, #72277873
maximal normal subgroup of	AGL(1,𝔽₇)×#3490, #72277632, #72277873

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