Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5925 |
| 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 | |
767 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 767 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)(3,15,20,13,6,31,8,27,18,5,19,32)(4,21,26,14,10,33,12,29,24,9,25,34)(7,16,17,28)(11,22,23,30);(3,4,8,12)(5,14,13,9)(6,10,19,25)(7,11,17,23)(15,29,27,21)(16,30,28,22)(18,24,20,26)(31,34,32,33);(5,13)(6,7)(9,14)(10,11)(15,28)(16,27)(17,19)(18,20)(21,30)(22,29)(23,25)(24,26);(1,2)46 conjugacy classes — view the full character table.
| maximal subgroup of | #11712948, #11712956, #11712963, … |
| maximal normal subgroup of | #11712948, #11712956, #11712963, … |
| maximal quotient of | C2×#85528, C2×#85366, C2×#85365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20068);