Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C23×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Fitting | #585 | D3 |
| hypercenter | #41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ #41 | |
759 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 759 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:
(1,25,15,11,16,28,6,12,17,3,4,27)(2,29,23,18,19,32,7,21,24,8,9,31)(5,13,14,26)(10,22,20,30);(2,7)(8,18)(9,19)(10,20)(21,29)(22,30)(23,24)(31,32);(1,2)(3,8)(4,9)(5,10)(6,7)(11,18)(12,21)(13,22)(14,20)(15,23)(16,19)(17,24)(25,29)(26,30)(27,31)(28,32);(3,11)(4,14)(5,16)(8,18)(9,20)(10,19)(12,13)(15,17)(21,22)(23,24)(25,26)(29,30)Polycyclic presentation:
2^4,3,2^3|d^2=h,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 | #11713196, #11713198, #11713427, … |
| maximal normal subgroup of | #11713196, #11713198, #11713427, … |
| maximal quotient of | #11714609, #11714642, #11714646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20074);