#72229458

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₇×M₁₆ ⋊ C₂×C₃
Hall	#11735980 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3184
Sylow	C₇ ⋊ C₃×#3184

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₄⋊AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₄×C₇×M₁₆	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×D₄
hypercenter	C₄×M₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3184
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3184, #72259679, #72259710
maximal normal subgroup of	AGL(1,𝔽₇)×#3184, #72259679, #72259710

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