Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | #519 | D3 |
| prime radical | C22×SL(2,𝔽3) | C22 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C22⋊C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
257 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 257 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)(6,21,28)(7,16,34,15,29,32)(8,19,10)(11,25,36,24,33,35)(12,27,14)(17,30)(26,31);(5,7)(6,11)(8,16)(9,17)(10,15)(12,25)(13,26)(14,24)(18,31)(19,32)(20,29)(21,33)(22,34)(23,30)(27,35)(28,36);(5,6,10,14)(7,11,15,24)(8,18,20,13)(9,21,23,12)(16,31,29,26)(17,33,30,25)(19,28,22,27)(32,36,34,35)Polycyclic presentation:
2^4,3,2^3|a^2=h,c^2=d,f^2=h,g^2=h,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 | #11712947, C2×#85350, #11714641, … |
| maximal normal subgroup of | #11712947, C2×#85350, #11714641, … |
| maximal quotient of | #11711837, #11711853, #11711873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17926);