Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#5429 |
| Fitting | C2×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×H(5,𝔽2) | D3 |
| hypercenter | C2×Q8 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×Q8 | |
345 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 3 generators on 26 points:
(3,17,26,8,13,23,9,15,19,4,21,22)(5,18,20,10,16,14,12,11,25,7,6,24);(4,10)(5,12)(6,14)(7,8)(11,19)(13,22)(15,20)(16,24)(17,25)(18,26)(21,23);(1,2)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5429, #11712832, #11713087, … |
| maximal normal subgroup of | C4×#5429, #11712832, #11713087, … |
| maximal quotient of | C2×#73092, C2×#73090, C2×#73100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18101);