Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#11711638 |
| Fitting | C2×#3027 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #73081 |
| commutator, cosocle | #5950 | C23 |
| Frattini | C22×C4 | C22×S4 |
| Fitting | C2×#3027 | D3 |
| hypercenter | C2×C4 | #1020⋊C2 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ #5950 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1563 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 1563 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2)(3,21,7)(4,17,12)(5,33,16)(8,19,20)(9,31,23)(10,34,25)(13,26,28)(14,32,30);(4,9)(5,14)(6,18)(7,20)(10,13)(11,24)(12,25)(17,34)(19,21)(22,27)(23,28)(26,31);(3,4)(5,9)(6,17)(7,12)(8,13)(10,14)(11,21)(15,31)(16,23)(18,26)(19,27)(20,28)(22,33)(24,32)(25,30)(29,34)50 conjugacy classes — view the full character table.
| maximal subgroup of | #72341742, #72341747, C2×#72236850, … |
| maximal normal subgroup of | #72341742, #72341747, C2×#72236850, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,21,7)(4,17,12)(5,33,16)(8,19,20)(9,31,23)(10,34,25)(13,26,28)(14,32,30), (4,9)(5,14)(6,18)(7,20)(10,13)(11,24)(12,25)(17,34)(19,21)(22,27)(23,28)(26,31), (3,4)(5,9)(6,17)(7,12)(8,13)(10,14)(11,21)(15,31)(16,23)(18,26)(19,27)(20,28)(22,33)(24,32)(25,30)(29,34));