Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C23×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Fitting | C2×2−1+4 | D3 |
| hypercenter | C2×D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
738 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 4 generators on 32 points:
(1,16,18)(2,9,24,8,17,23)(3,25,30)(4,15,6)(7,21,32,20,29,31)(10,14)(11,12,28)(22,27);(3,11)(7,20)(12,25)(13,26)(21,29)(22,27)(28,30)(31,32);(1,3)(2,7)(4,12)(5,13)(6,11)(8,20)(9,21)(10,22)(14,27)(15,28)(16,25)(17,29)(18,30)(19,26)(23,31)(24,32);(1,2)(3,7)(4,14)(5,17)(6,8)(9,19)(10,16)(11,20)(12,27)(13,29)(15,24)(18,23)(21,26)(22,25)(28,32)(30,31)Polycyclic presentation:
2^4,3,2^3|f^2=h,g^2=h,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #11713173, #11713175, #11713186, … |
| maximal normal subgroup of | #11713173, #11713175, #11713186, … |
| maximal quotient of | #11714608, #11714643, #11714645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20072);