#72277370

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

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₇×#62817 ⋊ C₃
derived	C₇×H(3,𝔽₄) ⋊ C₂²×C₃
Sylow	#62817 ⋊ ΓL(1,𝔽₈)
Sylow	C₇ ⋊ #11710731

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×H(3,𝔽₄)	C₂²×C₃
Frattini	C₂²	C₂²×C₂²⋊(C₂²⋊ΓL(1,𝔽₈))
Fitting	C₇×#62817	C₃
socle	C₂²×C₇	C₂²×C₂²⋊A₄
LCS	G ▷ C₇×H(3,𝔽₄)
UCS	C₁

3368 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 3368 conjugacy classes of subgroups — too many to draw. Only its 77 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#11710731
maximal normal subgroup of	ΓL(1,𝔽₈)×#11710731

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