Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | D4×#5935 |
| Fitting | D4×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×#1020 |
| commutator, cosocle | C2×#997 | C23 |
| Fitting | D4×H(5,𝔽2) | D3 |
| hypercenter | C2×D4 | #1020 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
3751 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 3751 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) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(5,11,12,9,10,7,8,6);(2,4)(6,7,12)(8,9,11);(1,2,3,4)65 conjugacy classes — view the full character table.
| maximal subgroup of | D4×SO+(4,𝔽3), D4×#73026, #72246076, … |
| maximal normal subgroup of | D4×#73026, #72246076, #72246231, … |
| maximal quotient of | C2×D4×#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,11,12,9,10,7,8,6), (2,4)(6,7,12)(8,9,11), (1,2,3,4));