Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #585 | D3 |
| hypercenter | #41 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
306 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(1,21,18,2,17,26)(3,19,30,12,27,31)(4,24,6,9,16,8)(5,10)(7,13,32,11,25,29)(14,23)(15,22)(20,28);(2,8)(7,12)(9,21)(10,22)(19,25)(20,23)(24,26)(31,32);(2,7)(3,11)(4,15)(5,17)(8,12)(9,23)(10,25)(13,14)(16,18)(19,22)(20,21)(24,32)(26,31)(27,28)Polycyclic presentation:
2^4,3,2^3|d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,d^c=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11713427, #11713428, #11713429, … |
| maximal normal subgroup of | #11713427, #11713428, #11713429, … |
| maximal quotient of | #11712086, #11712087, #11712088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18119);