#72276879

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#354 ⋊ C₂×C₃
Hall	#11726908 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #2199
Sylow	C₇ ⋊ C₃×#2199

Subgroups

	subgroup	G/H
center	C₂	#89513⋊C₃
commutator, cosocle	C₂×C₇×D₄	C₂³×C₃
Frattini	C₂×D₄	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#354	C₂×C₃
socle	C₂×C₇	C₃×B(4,𝔽₂)
hypercenter	#354	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#2199
LCS	G ▷ C₂×C₇×D₄ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #354

540 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^3,3,2^4,7|a^2=g,b^2=ef,c^2=h,e^2=h,b^a=be,c^a=cegh,c^b=cefh,e^b=eg,f^a=fgh,f^b=fg,f^e=fh,g^b=gh,g^c=gh,i^c=i^6,i^d=i^2

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#2199
maximal normal subgroup of	AGL(1,𝔽₇)×#2199
maximal quotient of	#72277290

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