#72219864

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

derived

#997⋊C₉ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72213207
commutator, cosocle, hypocenter	#997⋊C₉	C₂×C₃
Frattini, socle	C₂×C₃	C₃×A₄⋊S₄
Fitting	C₃²×H(5,𝔽₂)	C₃⋊D₃
LCS	G ▷ #997⋊C₉
UCS	C₁ ◁ C₂

527 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^4,2^5|d^3=e,f^2=j,g^2=j,c^a=c^2,d^a=d^2e^2,d^b=de,e^a=e^2,f^a=fj,f^c=g,f^d=g,g^a=fg,g^c=fgj,g^d=fgj,g^f=gj,h^a=ghij,h^c=i,h^d=fgh,h^g=hj,i^a=fij,i^c=hi,i^d=fij,i^f=ij

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72219864, #72245689, #72245730, …
maximal normal subgroup of	C₂×#72219864, #72245689, #72245730, …
maximal quotient of	C₂×#72219864, #72252975
holomorph of	Q₈⋊C₉

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