Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | 2−1+6 ⋊ C3×D3 |
| derived | #87537 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | S4×C22⋊A4 |
| commutator, cosocle, hypocenter | #87537 | C2×C3 |
| Fitting | 2−1+6 | C3×D3 |
| LCS | G ▷ #87537 | |
| UCS | C1 ◁ C2 | |
1968 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,15,4,5,6,16,14,17)(2,31,12,10,18,28,21,30,3,23,19,13,8,32,25,22,7,27,9,20,11,24,29,26);(2,8)(4,5,16)(7,18)(9,22,24,21,10,23)(12,13,28)(14,17,15)(19,30,32,29,20,31)(25,26,27);(1,7)(2,3)(4,19)(5,20)(6,18)(8,11)(9,12)(10,13)(14,29)(15,31)(16,32)(17,30)(21,25)(22,26)(23,27)(24,28)Polycyclic presentation:
2,3,2^4,3,2^3|h^2=j,i^2=j,c^b=d,d^b=cd,e^b=ef,e^c=ej,f^b=e,f^d=fj,g^a=g^2,h^a=hj,h^g=i,i^a=hi,i^g=hij,i^h=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72243002 |
| maximal normal subgroup of | #72243002 |
| maximal quotient of | GL(2,𝔽3)×#997 |
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, 3, 2, 2, 2, 2, 3, 2, 2, 2]);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);