Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | C4×#41 | D3 |
| prime radical | C2×#986 | C2 |
| hypercenter | C42 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C42 | |
271 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 38 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)(5,19,24)(6,13,28,12,23,27)(7,31,34)(8,9,22)(11,25,36,15,29,35)(14,21)(16,17,33)(26,30);(5,7,8,16)(6,11,12,15)(9,17,19,31)(10,18,20,32)(13,25,23,29)(14,26,21,30)(22,33,24,34)(27,35,28,36);(2,4)(5,6)(7,15)(8,12)(9,21)(10,23)(11,16)(13,20)(14,19)(17,26)(18,25)(22,28)(24,27)(29,32)(30,31)(33,35)(34,36)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=dh,f^2=h,g^2=h,b^a=bh,c^a=cd,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11711216, #11713429, #11713433, … |
| maximal normal subgroup of | #11711216, #11713429, #11713433, … |
| maximal quotient of | #11712203, #11712204, #11712205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17923);