Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
| direct product | C2×#72224760 |
| Fitting | C2×#63011 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72224760 |
| commutator, cosocle | #3549⋊C3 | C23 |
| Frattini | C23 | C22×#1020 |
| Fitting | C2×#63011 | D3 |
| socle | C23 | #87524 |
| hypocenter | #5464 | C2×D4 |
| LCS | G ▷ #3549⋊C3 ▷ #5464 | |
| UCS | C1 ◁ C2 | |
10881 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 10881 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,5,3,7)(2,11,6,8)(4,19)(9,12)(10,21,13,14)(15,22)(16,23,18,20)(17,24);(1,17,21,8,4,14)(2,15,22,5,9,12)(3,13,24,11,10,19)(6,16,20,7,18,23);(1,2)(3,7)(4,9)(5,8)(6,11)(10,16)(12,21)(13,18)(14,22)(15,17)(19,23)(20,24)66 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#72224760, C22×#72224760 |
| maximal normal subgroup of | C4×#72224760, C22×#72224760 |
| maximal quotient of | #72232235, C2×#72260915, #72261375, … |
| automorphism group of | #1747, #2017, #2091, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,5,3,7)(2,11,6,8)(4,19)(9,12)(10,21,13,14)(15,22)(16,23,18,20)(17,24), (1,17,21,8,4,14)(2,15,22,5,9,12)(3,13,24,11,10,19)(6,16,20,7,18,23), (1,2)(3,7)(4,9)(5,8)(6,11)(10,16)(12,21)(13,18)(14,22)(15,17)(19,23)(20,24));