Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4621 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #857 |
| commutator, prime radical, hypocenter | #95⋊C3 | C4 |
| Frattini | C22×C4 | S4 |
| Fitting | #376 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| cosocle | #376⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,11,6,3)(2,9)(4,21,15,23,16,13,17,12)(5,7)(8,22,14,18,19,20,10,24);(1,4,13)(2,8,12)(3,10,18)(5,15,22)(6,16,21)(7,17,20)(9,19,23)(11,14,24)Polycyclic presentation:
2^2,3,2^5|a^2=b,b^2=h,d^2=f,e^2=fg,c^a=c^2,d^a=ef,e^a=dfg,f^a=fg,d^c=efh,e^c=deh,f^c=fg,g^c=fh,e^d=eh30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67997, #11710982, #11711007, … |
| maximal normal subgroup of | C2×#67997, #11710982, #11711007, … |
| maximal quotient of | #11710943, C2×#67997, #11919925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 573);