Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×#984 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini, socle | C22 | C22×S4 |
| Fitting | C4×#41 | D3 |
| prime radical | C2×#41⋊C3 | C22 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C42 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
260 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,16,22,18,8,32,10,33,20,7,21,34)(6,23,28,15,12,35,14,29,26,11,27,36)(9,17,19,31)(13,24,25,30);(5,6,10,14)(7,15,18,11)(8,13,21,25)(9,12,19,27)(16,30,33,24)(17,29,31,23)(20,28,22,26)(32,35,34,36);(1,2,3,4)52 conjugacy classes — view the full character table.
| maximal subgroup of | #11711108, #11711151, C8×#984, … |
| maximal normal subgroup of | #11711108, #11711151, C8×#984, … |
| maximal quotient of | C8×#984, #11711848, #11711864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17909);