Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Fitting | C4×#41 | D3 |
| socle | C22 | C2×SL(2,ℤ/4) |
| prime radical, cosocle | C2×#41⋊C3 | C22 |
| hypercenter | C42 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,20,22)(6,13,30,11,21,28)(7,29,34)(8,19,10)(12,25,36,24,33,35)(14,18)(15,32,17)(26,31);(5,6,7,12,10,11,17,24)(8,18,15,31,20,14,29,26)(9,21,16,33,23,13,27,25)(19,30,32,36,22,28,34,35)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=d,c^2=h,f^2=h,g^2=h,b^a=bh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #11711037, #11711038, #11711166, … |
| maximal normal subgroup of | #11711037, #11711038, #11711166, … |
| maximal quotient of | #11710987, #11710989, #11710995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5643);