Finite Group DatabasebetaFinite Group Databasebeta| order | 304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C19 ⋊ C2 |
| Sylow | C19 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D19 |
| commutator, socle, cosocle | C2×C19 | C23 |
| Fitting | Q8×C19 | C2 |
| hypercenter | Q8 | D19 |
| hypocenter | C19 | #41 |
| LCS | G ▷ C2×C19 ▷ C19 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 152 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,22,23,26)(21,24,25,27);(20,21,23,25)(22,27,26,24);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(22,26)(24,27)Polycyclic presentation:
2^4,19|b^2=d,c^2=d,c^a=cd,e^a=e^18,c^b=cd55 conjugacy classes — view the full character table.
| maximal subgroup of | #10597360, #10597362, #10597364, … |
| maximal normal subgroup of | #10597360, #10597362, #10597364, … |
| maximal quotient of | #10597336, #10597338, #10597339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(304, 34);