Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#11711590 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C22×C4 | C22×S4 |
| Fitting | C2×#3025 | D3 |
| socle | C22 | #73080 |
| prime radical | C2×#3025⋊C3 | C2 |
| cosocle | #5950 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
1241 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1241 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(3,7,28)(4,21,31,9,12,24)(5,14,33)(6,16)(8,18,17)(10,30,34,19,23,32)(13,25)(15,27,26);(3,4,5,10,8,9,15,19)(6,12,13,23,16,21,25,30)(7,11,14,22,18,20,27,29)(17,24,26,32,28,31,33,34);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236801, C2×#72236787, C2×#72236805, … |
| maximal normal subgroup of | C2×#72236801, C2×#72236787, C2×#72236805, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,7,28)(4,21,31,9,12,24)(5,14,33)(6,16)(8,18,17)(10,30,34,19,23,32)(13,25)(15,27,26), (3,4,5,10,8,9,15,19)(6,12,13,23,16,21,25,30)(7,11,14,22,18,20,27,29)(17,24,26,32,28,31,33,34), (1,2));