Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#11714177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | C2×#3025 | D3 |
| prime radical | C2×#3025⋊C3 | C2 |
| hypercenter | C2×C4 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
3181 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 3181 conjugacy classes of subgroups — too many to draw. Only its 107 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 — 4 generators on 34 points:
(3,15,22,5,20,32,8,31,19,16,7,30)(4,24,26,9,18,34,12,28,25,13,11,33)(6,29,17,14)(10,27,21,23);(3,4,8,12)(5,13,16,9)(6,18,17,11)(7,21,20,10)(14,24,29,28)(15,23,31,27)(19,26,22,25)(30,33,32,34);(6,17)(7,20)(10,21)(11,18)(14,29)(15,31)(19,22)(23,27)(24,28)(25,26)(30,32)(33,34);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236914, C2×#72236916, C2×#72236883, … |
| maximal normal subgroup of | C2×#72236914, C2×#72236916, C2×#72236883, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,15,22,5,20,32,8,31,19,16,7,30)(4,24,26,9,18,34,12,28,25,13,11,33)(6,29,17,14)(10,27,21,23), (3,4,8,12)(5,13,16,9)(6,18,17,11)(7,21,20,10)(14,24,29,28)(15,23,31,27)(19,26,22,25)(30,33,32,34), (6,17)(7,20)(10,21)(11,18)(14,29)(15,31)(19,22)(23,27)(24,28)(25,26)(30,32)(33,34), (1,2));