#72268822

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₉×#2021 ⋊ C₂
Sylow	C₉ ⋊ #23900

Subgroups

	subgroup	G/H
center	C₂²	D₉×#120
commutator, cosocle	C₂⁴×C₉	C₂⁴
Frattini	C₂⁴×C₃	C₂³×D₃
Fitting	C₉×#2021	C₂
socle	C₂²×C₃	D₃×#120
hypercenter	#2021	D₉
hypocenter	C₉	#23900
LCS	G ▷ C₂⁴×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2021

5280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 21 points:

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

Polycyclic presentation:

2^8,3^2|i^3=j,b^a=bf,c^a=ce,d^c=dg,e^d=eh,g^a=gh,i^b=i^2j^2,i^d=i^2j^2,j^b=j^2,j^d=j^2

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#23900
maximal normal subgroup of	D₉×#23900

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