#72259693

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇×Q₈ ⋊ C₂×C₃²
Hall	#6551 ⋊ C₃²
Sylow	C₇ ⋊ C₃×2₋¹⁺⁴⋊C₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×A₄×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇×Q₈	C₂²×C₃²
Fitting	C₂×C₇×Q₈	C₂×C₃²
socle	C₂×C₇	C₂²×C₃×A₄
hypercenter	C₂²	A₄×AGL(1,𝔽₇)
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂ ◁ C₂²

236 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 112

Polycyclic presentation:

2^2,3^2,2^2,7,2|e^2=h,f^2=h,b^a=bh,e^d=f,f^d=efh,f^e=fh,g^b=g^6,g^c=g^2

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72259707

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