Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #520 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | #520 | D3 |
| prime radical | C2×#985 | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C4⋊C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
265 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,3,4)(5,20,22)(6,13,28,12,21,27)(7,16,34,15,29,32)(8,19,10)(11,33,36)(14,18)(17,30)(24,35,26);(5,7,10,15)(6,11,12,26)(8,16,20,29)(9,17,23,30)(13,24,21,33)(14,25,18,31)(19,32,22,34)(27,35,28,36);(5,6)(7,11)(8,18)(9,21)(10,12)(13,23)(14,20)(15,26)(16,31)(17,33)(19,28)(22,27)(24,30)(25,29)(32,36)(34,35)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=d,f^2=h,g^2=h,c^a=ch,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 | #11712958, #11712959, #11712963, … |
| maximal normal subgroup of | #11712958, #11712959, #11712963, … |
| maximal quotient of | #11711838, #11711854, #11711874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17941);