#72259691

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#390 ⋊ C₂×C₃
Hall	#11733759 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #2433
Sylow	C₇ ⋊ C₃×#2433

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂³×C₇	C₂⁴×C₃
Fitting	C₇×#390	C₂×C₃
hypercenter	#390	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#2433
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #390

2696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 19 points:

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

Polycyclic presentation:

2^4,3,2^3,7|a^2=f,d^2=gh,c^a=cf,d^b=dg,d^c=dh,i^b=i^6,i^e=i^4

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

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