Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | C22×#41 | D3 |
| hypercenter | C22×C4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
311 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 311 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,20,22)(6,13,28,12,21,27)(7,31,33)(8,19,10)(11,25,36,24,32,35)(14,18)(15,30,17)(26,29);(2,4)(5,7,10,17)(6,11,12,24)(8,15,20,31)(9,16,23,34)(13,25,21,32)(14,26,18,29)(19,30,22,33)(27,35,28,36);(1,2)(3,4)(5,6)(7,11)(8,18)(9,21)(10,12)(13,23)(14,20)(15,29)(16,32)(17,24)(19,28)(22,27)(25,34)(26,31)(30,36)(33,35)Polycyclic presentation:
2^4,3,2^3|b^2=h,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11713186, C2×#85522, #11714682, … |
| maximal normal subgroup of | #11713186, C2×#85522, #11714682, … |
| maximal quotient of | #11711987, #11711991, #11711994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18098);