#72253027

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×C₃×C₂²⋊C₄ ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×S₄
commutator, cosocle	C₂²×C₃×A₄	C₂⁴
Fitting	C₂²×C₃×C₂²⋊C₄	C₂×D₃
socle	C₂⁴×C₃	C₂³×D₃
hypercenter	C₂²⋊C₄	D₃×S₄
hypocenter	C₃×A₄	D₄²
LCS	G ▷ C₂²×C₃×A₄ ▷ C₃×A₄
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

5398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 15 points:

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

Polycyclic presentation:

2^6,3^2,2^2|d^2=f,b^a=be,d^b=df,d^c=df,g^a=g^2,g^d=g^2,h^c=h^2,i^a=j,i^d=j,i^g=j,j^a=i,j^d=i,j^g=ij

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11714540
maximal normal subgroup of	D₃×#11714540

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