Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting, Sylow | #2276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C42⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2276 | C3 |
| prime radical | #95⋊C3 | C22 |
| hypercenter | Q8 | C42⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,6,4,5,9)(3,7,11,8,10,12)(13,23,34,18,16,31,19,27,24,15,25,36)(14,26,32,22,20,35,21,29,28,17,30,33);(1,3,4,8)(2,10,5,7)(15,18)(16,23)(17,22)(20,29)(24,33)(25,27)(26,30)(28,36)(31,32)(34,35)Polycyclic presentation:
2^2,3,2^5|a^2=f,b^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cfgh,c^b=cfh,d^c=deg,e^c=dgh,g^c=fh,h^c=gh,e^d=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #20506⋊C3, #20507⋊C3, #33089⋊C3, … |
| maximal normal subgroup of | #20506⋊C3, #20507⋊C3, #33089⋊C3, … |
| maximal quotient of | #13750⋊C3, C4⋊(#376⋊C3), Q8×#95⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5742);