#72241537

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₃² ⋊ C₄×D₃
derived	C₃²×C₂²⋊A₄ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₃²×C₂²⋊A₄	C₂×C₄
Frattini	C₂²	S₄×#157
Fitting	C₂⁴×C₃²	C₄×D₃
socle	C₂²×C₃²	C₄×S₄
prime radical	C₃⋊D₃×#1020	C₂
cosocle	C₃⋊D₃×C₂²⋊A₄	C₂²
LCS	G ▷ C₃²×C₂²⋊A₄
UCS	C₁

1002 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 14
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 14 points:

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

Polycyclic presentation:

2^3,3^3,2^4|b^2=c,d^a=d^2,e^b=ef,e^c=e^2,f^b=ef^2,f^c=f^2,g^b=gj,g^d=h,h^a=gh,h^b=hi,h^d=gh,i^a=ij,i^d=ij,j^d=i

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241545
maximal normal subgroup of	#72241545

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