Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×M64 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#73035 |
| commutator, cosocle | A4×C16 | C23 |
| Frattini | C16 | C22×S4 |
| Fitting | C22×M64 | D3 |
| socle | C23 | C2×D48 |
| hypercenter | M64 | S4 |
| hypocenter | A4 | #2263 |
| LCS | G ▷ A4×C16 ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ M64 | |
503 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,4,3)(6,11)(12,24)(13,25)(14,23)(15,27)(20,35)(26,28)(29,34);(2,4)(5,6)(7,15)(8,20)(9,23)(10,11)(12,17)(13,16)(14,21)(18,34)(19,27)(22,35)(24,33)(25,31)(26,36)(28,30)(29,32);(5,6,7,12,16,20,30,29,9,14,18,26,22,25,33,27,10,11,19,24,31,35,36,34,21,23,32,28,8,13,17,15)Polycyclic presentation:
2^7,3,2^2|a^2=d,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,c^b=cg,d^b=de,e^b=ef,f^b=fg,h^b=h^2,i^b=ij,i^h=j,j^h=ij92 conjugacy classes — view the full character table.
| maximal subgroup of | #72342827, S4×#2263, #72342910 |
| maximal normal subgroup of | #72342827, S4×#2263, #72342910 |
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, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);