Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #568 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | #568 | D3 |
| socle | C2 | C2×C4×S4 |
| prime radical | C2×#985 | C2 |
| cosocle | #41⋊C3 | C23 |
| hypercenter | M16 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ M16 | |
250 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,19,28,8,4,23,13,21,17,2,27,32,6,9,26,18,16,24,3,29,15,7,11,31)(5,10,12,22,14,20,25,30);(2,7)(8,18)(9,19)(10,20)(21,29)(22,30)(23,24)(31,32);(4,14)(5,16)(9,20)(10,19)(11,25)(12,27)(15,17)(21,30)(22,29)(23,24)(26,28)(31,32)Polycyclic presentation:
2^4,3,2^3|c^2=d,d^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #11712948, #11712949, #11712950, … |
| maximal normal subgroup of | #11712948, #11712949, #11712950, … |
| maximal quotient of | #11711836, #11711837, #11711838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18028);