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 | #520 | D3 |
| prime radical | C2×#41⋊C3 | C22 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| hypercenter | C4⋊C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
255 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 255 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,3,4,2)(5,18,23)(6,22,28)(7,16,34,15,29,32)(8,9,21)(11,25,36,24,33,35)(12,13,27)(17,30)(26,31);(5,7,8,15)(6,11,12,24)(9,16,18,29)(10,17,19,30)(13,25,22,33)(14,26,20,31)(21,32,23,34)(27,35,28,36);(1,2,4,3)(5,6,8,12)(7,11,15,24)(9,20,18,14)(10,22,19,13)(16,31,29,26)(17,33,30,25)(21,28,23,27)(32,36,34,35)Polycyclic presentation:
2^4,3,2^3|a^2=d,b^2=h,c^2=dh,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 | #11713086, #11713088, #11713090, … |
| maximal normal subgroup of | #11713086, #11713088, #11713090, … |
| maximal quotient of | #11711835, #11711851, #11711871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17936);