Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | C22×#131 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | C22⋊A4 |
| commutator, hypocenter | C22×Q8 | C3×C4 |
| Frattini | C4 | C2×C22⋊A4 |
| Fitting | C22×#131 | C3 |
| socle | C23 | C4×A4 |
| prime radical | #5948 | C2 |
| cosocle | C22×#41 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C8 | |
289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4,3)(5,6,7,11,10,14,17,24)(8,25,32,22,19,13,31,34,21,12,29,36,20,26,16,35,18,28,15,33,9,27,30,23);(1,4)(2,3);(5,19,10,18)(6,26,14,25)(7,30,17,29)(8,21,20,9)(11,34,24,33)(12,28,27,13)(15,32,31,16)(22,36,35,23)Polycyclic presentation:
2,3,2^6|a^2=c,c^2=h,d^2=h,e^2=h,d^b=deh,e^b=dh,f^b=fgh,g^b=f,e^d=eh48 conjugacy classes — view the full character table.
| maximal subgroup of | #12680⋊C3, #21442⋊C3, #21444⋊C3, … |
| maximal normal subgroup of | #12680⋊C3, #21442⋊C3, #21444⋊C3, … |
| maximal quotient of | C8×C22⋊SL(2,𝔽3), #131⋊SL(2,𝔽3), #60304⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18241);