#72276552

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×Q₈ ⋊ D₃²
derived	C₃×A₄×SL(2,𝔽₃) ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#13181790
commutator, cosocle, hypocenter	C₃×A₄×SL(2,𝔽₃)	C₂²
Fitting	C₂²×C₃×Q₈	D₃²
socle	C₂³×C₃	D₃×S₄
LCS	G ▷ C₃×A₄×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

982 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,4)(5,6)(8,14,13,10,11,9,15,12);(2,3)(6,7)(9,10,15)(12,13,14)

Polycyclic presentation:

2^2,3^3,2^5|h^2=j,i^2=j,c^a=c^2,c^b=c^2,d^b=d^2,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^b=ij,h^d=i,i^b=hj,i^d=hij,i^h=ij

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×S₄×GL(2,𝔽₃)
maximal normal subgroup of	D₃×S₄×GL(2,𝔽₃)

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