Finite Group Databasebeta

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#72241520

order3456
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23×C3

Aliases

No aliases for this group.

Splittings

derivedC3×A42 ⋊ C2×C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2S4×C3⋊S4
commutator, hypocenterC3×A42C2×C4
Fitting, socleC25×C3D32
prime radicalC2×C3⋊(A4⋊S4)C2
cosocleC2×C3×A42C22
LCSG ▷ C3×A42
UCSC1 ◁ C2

1410 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 1410 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 108

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#11923458, #72252927, #72252976
maximal normal subgroup ofD3×#11923458, #72252927, #72252976

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