#72276718

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#534 ⋊ C₂×C₃
Hall	#11743378 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3532
Sylow	C₇ ⋊ C₃×#3532

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁵×C₃
Fitting	C₇×#534	C₂×C₃
hypercenter	#534	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3532
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #534

2672 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)(3,4,6)(10,14)(12,15)(18,22)(21,23);(9,13)(10,14)(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,12,13,15)(16,17)(18,23)(19,20)(21,22);(10,14)(12,15)(17,20)(18,22);(2,3)(4,6)(5,7)(8,9)(10,12)(11,13)(14,15)(17,20)(18,22)

Polycyclic presentation:

2^5,3,2^2,7|c^2=h,d^2=g,c^a=cg,c^b=cgh,d^a=dgh,d^b=dg,d^c=dgh,e^c=eh,e^d=eg,i^a=i^6,i^f=i^4

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3532, #72277632
maximal normal subgroup of	AGL(1,𝔽₇)×#3532, #72277632

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