Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2634 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C4×Q8 | C2 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #2634 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
416 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(8,10,11,14)(9,13,12,15)(19,27)(22,29)(25,31)(28,30);(2,3)(5,7)(9,12)(13,15)(16,19,20,27)(17,22,23,29)(18,25,26,31)(21,28,24,30);(4,5)(6,7)(16,18,20,26)(17,21,23,24)(19,25,27,31)(22,28,29,30);(8,9)(10,13)(11,12)(14,15)(16,17,20,23)(18,24,26,21)(19,22,27,29)(25,30,31,28)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=e,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10691183, #10691185, #10696081, … |
| maximal normal subgroup of | #10691183, #10691185, #10696081, … |
| maximal quotient of | #10721686, #10721845, #10721925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15062);