Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5924 |
| Fitting | C2×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C2×H(5,𝔽2) | D3 |
| hypercenter | C2×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
737 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 737 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2)(3,14,20,16,6,30,8,31,18,5,19,32)(4,21,26,13,10,33,12,27,24,9,25,34)(7,15,17,29)(11,22,23,28);(3,4,8,12)(5,13,16,9)(6,10,19,25)(7,11,17,23)(14,27,31,21)(15,28,29,22)(18,24,20,26)(30,34,32,33);(6,7)(10,11)(14,15)(17,19)(18,20)(21,22)(23,25)(24,26)(27,28)(29,31)(30,32)(33,34);(1,2)46 conjugacy classes — view the full character table.
| maximal subgroup of | #11712950, #11712959, #11712966, … |
| maximal normal subgroup of | #11712950, #11712959, #11712966, … |
| maximal quotient of | C2×#85528, C2×#85360, C2×#85364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20067);