Finite Group DatabasebetaFinite Group Databasebeta| order | 2604 |
| isoclinism | isoclinic class of D651 |
| automorphism | Aut(G) order 468720 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D1302 |
| direct product | C2×D651 |
| unambiguous semidirect product | (C2×C3)⋊D217 |
| unambiguous semidirect product | (C2×C7)⋊D93 |
| unambiguous semidirect product | (C2×C31)⋊D21 |
| Fitting, socle | C2×C3×C7×C31 ⋊ C2 |
| derived, Hall | C3×C7×C31 ⋊ C22 |
| Hall | C7×C31 ⋊ C2×D3 |
| Hall | C3×C31 ⋊ C2×D7 |
| Sylow | C31 ⋊ C2×D21 |
| Hall | C3×C7 ⋊ C2×D31 |
| Sylow | C7 ⋊ C2×D93 |
| Sylow | C3 ⋊ C2×D217 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D651 |
| commutator, cosocle, hypocenter | C3×C7×C31 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7×C31 | C2 |
| LCS | G ▷ C3×C7×C31 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1302 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2)(3,4,5)(6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(4,5)(7,12)(8,11)(9,10)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)654 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C3⋊(C7⋊(D31⋊C5)), D5×D651, C2×D3255, … |
| maximal normal subgroup of | C2×C3⋊(C7⋊(D31⋊C5)), C2×C5×D651, C2×C7×D651, … |
| maximal quotient of | D5×D651, C2×D3255, C2×C5×D651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2604, 64);