#72231670

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×#120 ⋊ D₃
derived	C₃×C₂²⋊(C₂²⋊A₄) ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₂²⋊(C₂²⋊A₄)	C₂²
Frattini	C₂²	C₂×C₃⋊#1020
Fitting	C₂²×C₃×#120	D₃
socle	C₂⁴×C₃	C₂×S₄
LCS	G ▷ C₃×C₂²⋊(C₂²⋊A₄)
UCS	C₁

2288 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 2288 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 48

A minimal faithful permutation representation — 3 generators on 15 points:

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

Polycyclic presentation:

2^2,3^2,2^6|c^a=c^2,d^a=d^2,e^a=f,e^d=f,f^a=e,f^d=ef,g^a=h,g^b=gij,g^d=h,h^a=g,h^b=hi,h^d=gh,i^a=ij,i^d=j,j^d=ij

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#87588⋊C₂, C₂×#72231670
maximal normal subgroup of	D₃×#87588⋊C₂, C₂×#72231670
maximal quotient of	#72231672, C₂×#72231670

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