Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #68001 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625 |
| commutator, prime radical, hypocenter | C22⋊SL(2,𝔽3) | C4 |
| Frattini | C22×C4 | S4 |
| Fitting | #395 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| cosocle | #395⋊C3 | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3)(2,9)(4,21,16,22,15,13,14,12)(6,11)(8,23,20,24,18,19,10,17);(1,4,13)(2,8,12)(3,10,17)(5,14,23)(6,15,21)(7,16,19)(9,18,22)(11,20,24)Polycyclic presentation:
2^2,3,2^5|a^2=b,b^2=h,d^2=h,e^2=h,c^a=c^2,d^a=egh,e^a=dfh,f^a=fg,d^b=dfg,e^b=ef,d^c=e,e^c=deh,f^c=g,g^c=fg,e^d=eh21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#68002, #11711366, #11711373, … |
| maximal normal subgroup of | C2×#68002, #11711366, #11711373, … |
| maximal quotient of | #11710951, C2×#68002, #11919930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 578);