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 | #5923 | 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) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,32,41,15,23,21,20,16,46,3,25,44,8,30,33,13,18,39,4,31,48,14,5,36)(2,38,45,6,29,42,17,26,34,10,12,43,9,24,40,19,35,37,11,28,47,7,27,22);(2,9)(3,14)(11,17)(12,27)(13,15)(16,31)(21,39)(29,35)(30,32)(34,47)(36,44)(40,45);(1,2)(3,10)(4,11)(5,21)(6,15)(7,14)(8,9)(12,22)(13,19)(16,33)(17,20)(18,36)(23,44)(24,45)(25,39)(26,47)(27,43)(28,34)(29,37)(30,48)(31,41)(32,46)(35,42)(38,40)Polycyclic presentation:
2^4,3,2^3|c^2=d,d^2=h,f^2=h,g^2=h,b^a=bh,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 | #11713094, #11713095, #11713136, … |
| maximal normal subgroup of | #11713094, #11713095, #11713136, … |
| maximal quotient of | #11711833, #983⋊C8, #11711835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18029);