#72277326

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#417 ⋊ C₂×C₃
Hall	#11728059 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #1792
Sylow	C₇ ⋊ C₃×#1792

Subgroups

	subgroup	G/H
center	C₂²	#6500⋊C₃
commutator	C₂×C₄×C₇	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#417	C₂×C₃
socle	C₂²×C₇	C₃×#120
prime radical	C₂×#10621518	C₂
cosocle	C₂²×C₄×C₇	C₂³×C₃
hypercenter	#417	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#1792
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

1240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

2^3,3,2^4,7|a^2=f,b^2=h,e^2=h,b^a=be,c^a=cg,c^b=ch,e^a=eh,e^b=eh,e^c=eh,g^b=gh,i^c=i^6,i^d=i^2

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#1792
maximal normal subgroup of	AGL(1,𝔽₇)×#1792

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