#72277443

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

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#410 ⋊ C₂×C₃
Hall	#11736135 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3025
Sylow	C₇ ⋊ C₃×#3025

Subgroups

	subgroup	G/H
center	C₄	#6445⋊C₃
commutator, cosocle	C₂³×C₇	C₂⁴×C₃
Frattini	C₂³	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#410	C₂×C₃
socle	C₂×C₇	C₂×C₃×#120
hypercenter	#410	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3025
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

1532 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)(8,11,12,20)(9,15,14,22)(10,17,18,19)(13,21,16,23);(2,3)(4,6)(5,7)(9,14)(15,22);(8,10)(9,13,14,16)(11,19)(12,18)(15,23,22,21)(17,20);(8,9)(10,16)(11,15)(12,14)(13,18)(17,23)(19,21)(20,22)

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3025
maximal normal subgroup of	AGL(1,𝔽₇)×#3025

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, 2, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, 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);