Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#11714236 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85547 |
| commutator, cosocle, hypocenter | C22⋊(#95⋊C3) | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C23×#95 | D3 |
| socle | C26 | S4 |
| prime radical | C2×C22⋊(#95⋊C3) | C2 |
| LCS | G ▷ C22⋊(#95⋊C3) | |
| UCS | C1 ◁ C22 | |
2162 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 2162 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2)(3,14,17)(4,19,11)(5,12,7)(6,24,22)(8,21,10)(9,25,26)(13,23,16)(15,18,20);(3,10,16,9,7,4,6,15)(5,26)(8,12,25,22)(11,13,21,24)(14,20)(17,18,23,19)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231239, C2×#72238916, C2×#72231199, … |
| maximal normal subgroup of | #72231239, C2×#72231199 |
| maximal quotient of | #72261592 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,14,17)(4,19,11)(5,12,7)(6,24,22)(8,21,10)(9,25,26)(13,23,16)(15,18,20), (3,10,16,9,7,4,6,15)(5,26)(8,12,25,22)(11,13,21,24)(14,20)(17,18,23,19));