#72229499

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×#100 ⋊ C₂×C₃
Hall	#11736145 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3019
Sylow	C₇ ⋊ C₃×#3019

Subgroups

	subgroup	G/H
center	C₂	C₂×#6445⋊C₃
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₂×C₇×#100	C₂×C₃
socle	C₂×C₇	C₂×C₃×#120
hypercenter	C₂×#100	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3019
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

2^4,3,2^3,7|a^2=f,f^2=h,b^a=bg,c^a=cfh,d^a=dh,f^b=fh,f^c=fh,g^a=gh,g^c=gh,i^c=i^6,i^e=i^4

Conjugacy classes & character table

109 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, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);