Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×#994 |
| Fitting, Sylow | C4×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | C4×H(5,𝔽2) | C3 |
| prime radical | C2×#994 | C2 |
| cosocle | C2×Q8 | C23×C3 |
| hypercenter | C4×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
380 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,15,25,13,8,23,10,22,16,7,19,28)(6,20,24,9,12,27,14,18,21,11,17,26);(7,13)(8,18)(9,11)(12,22)(15,17)(16,28)(19,20)(21,27)(23,25)(24,26);(1,2,3,4)76 conjugacy classes — view the full character table.
| maximal subgroup of | #17438⋊C3, C4×#576⋊C3, C4×#577⋊C3, … |
| maximal normal subgroup of | #17438⋊C3, C4×#576⋊C3, C4×#577⋊C3, … |
| maximal quotient of | C4×#520⋊C3, C4×C4⋊(#41⋊C3), C4×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18156);