#72241578

order	3528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow	C₇² ⋊ C₃×#67
Hall	ΓL(1,𝔽₈)² ⋊ D₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	#64425	C₂²×C₃
Fitting, socle	C₇²	C₃×#67
hypocenter	#3862	C₃×D₄
LCS	G ▷ #64425 ▷ #3862
UCS	C₁

114 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 14

Polycyclic presentation:

2^2,3,2,3,7^2|b^a=bd,e^a=e^2,f^a=f^2g^3,f^b=f^6g^3,f^c=f^4,f^d=f^6,f^e=f^4g^4,g^a=f^6g^5,g^c=g^4,g^d=g^6,g^e=g^2

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

automorphism group of

#3862, D₇²

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