#72277323

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#3532 ⋊ C₂×C₃
derived	C₇×H(3,𝔽₄) ⋊ C₂²×C₃
Hall	D₇×#3532 ⋊ C₃
Sylow	C₇ ⋊ C₂×#3532⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×H(3,𝔽₄)	C₂²×C₃
Frattini	C₂²	C₂×D₇⋊(C₂²⋊A₄)
Fitting	C₇×#3532	C₂×C₃
socle	C₂²×C₇	C₂²×C₂²⋊A₄
LCS	G ▷ C₇×H(3,𝔽₄)
UCS	C₁

3394 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 3394 conjugacy classes of subgroups — too many to draw. Only its 57 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:

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

Polycyclic presentation:

2^2,3,2^4,7,2^2|d^2=i,e^2=ij,d^b=di,d^c=ei,e^b=eij,e^c=de,f^c=g,f^d=fj,f^e=fij,g^c=fg,g^d=gij,g^e=gi,h^a=h^6,h^c=h^4,i^c=ij,j^c=i

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3532⋊C₃
maximal normal subgroup of	AGL(1,𝔽₇)×#3532⋊C₃

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 7, 2, 2]);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);