Finite Group DatabasebetaFinite Group Databasebeta| order | 304 |
| isoclinism | isoclinic class of #64525 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C19 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D19 |
| commutator, cosocle | C4×C19 | C22 |
| Frattini | C4 | C2×D19 |
| Fitting, prime radical | D4×C19 | C2 |
| socle | C2×C19 | D4 |
| hypercenter | D4 | D19 |
| hypocenter | C19 | SD16 |
| LCS | G ▷ C4×C19 ▷ C2×C19 ▷ C19 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 152 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(21,24)(22,27)(25,26);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21,23,26)(22,24,27,25)Polycyclic presentation:
2^4,19|a^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^18,c^b=cd52 conjugacy classes — view the full character table.
| maximal subgroup of | #10597357, #10597358, SD16×D19, … |
| maximal normal subgroup of | #10597357, #10597358, SD16×D19, … |
| maximal quotient of | #10597265, #10597267, #10597289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(304, 15);