Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11715736 |
| Fitting | C2×#3573 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C4⋊#1020 |
| commutator, cosocle | #5950 | C23 |
| Frattini | C4 | C22×#1020 |
| Fitting | C2×#3573 | D3 |
| hypercenter | C2×C8 | #1020 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ #5950 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 | |
2974 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 2974 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(3,4,10,21,8,13,5,9)(6,26,32,30,7,33,14,31,16,11,15,34,17,12,27,19,18,24,22,20,28,25,23,29);(1,2)(3,6)(4,11)(5,14)(7,16)(8,17)(9,19)(10,22)(12,24)(13,25)(15,27)(18,28)(20,29)(21,30)(23,32)(26,33)(31,34);(4,9)(5,10)(6,7)(11,20)(12,19)(13,21)(14,23)(15,22)(17,18)(24,29)(25,31)(26,30)(27,32)(33,34)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341764, C2×#72236850, #72341790, … |
| maximal normal subgroup of | #72341764, C2×#72236850, #72341790, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,10,21,8,13,5,9)(6,26,32,30,7,33,14,31,16,11,15,34,17,12,27,19,18,24,22,20,28,25,23,29), (1,2)(3,6)(4,11)(5,14)(7,16)(8,17)(9,19)(10,22)(12,24)(13,25)(15,27)(18,28)(20,29)(21,30)(23,32)(26,33)(31,34), (4,9)(5,10)(6,7)(11,20)(12,19)(13,21)(14,23)(15,22)(17,18)(24,29)(25,31)(26,30)(27,32)(33,34));