#72259948

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×H(3,𝔽₄) ⋊ C₂×C₃
Hall	#11744158 ⋊ 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×H(3,𝔽₄)	C₂×C₃
hypercenter	H(3,𝔽₄)	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3532
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

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

Polycyclic presentation:

2^5,3,2^2,7|b^2=g,c^2=h,b^a=bg,c^a=ch,d^b=dgh,d^c=dh,e^b=eg,e^c=egh,i^a=i^6,i^f=i^4

Conjugacy classes & character table

133 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260001, AGL(1,𝔽₇)×#3532
maximal normal subgroup of	#72260001, AGL(1,𝔽₇)×#3532

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