#72259848

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₇×H(3,𝔽₄) ⋊ C₂×C₃
Hall	D₇×H(3,𝔽₄) ⋊ C₃
Sylow	C₇ ⋊ C₂×#5464

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×H(3,𝔽₄)	C₂×C₃
Frattini	C₂²	D₇⋊(C₂²⋊A₄)
socle	C₂²×C₇	C₂×C₂²⋊A₄
LCS	G ▷ C₇×H(3,𝔽₄)
UCS	C₁

576 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 26 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:

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

Polycyclic presentation:

2,3,2^4,7,2^2|c^2=h,d^2=i,c^b=dh,d^b=cd,e^b=ef,e^c=ehi,e^d=ei,f^b=e,f^c=fh,f^d=fhi,g^a=g^6,g^b=g^4,h^b=i,i^b=hi

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#5464, #72277323
maximal normal subgroup of	AGL(1,𝔽₇)×#5464, #72277323

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