Finite Group DatabasebetaFinite Group Databasebeta| order | 30282 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Hall | C49×C103 ⋊ C2×C3 |
| Hall | #29502872 ⋊ C2 |
| Hall | D5047 ⋊ C3 |
| Sylow | C103 ⋊ D49⋊C3 |
| Sylow | C49 ⋊ D103⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C103 | C2×C3 |
| Frattini | C7 | #16864505 |
| socle | C7×C103 | AGL(1,𝔽7) |
| LCS | G ▷ C49×C103 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 152 |
| transitive degree | μt(G) = 5047 |
A minimal faithful permutation representation — 2 generators on 152 points:
(2,10,12,4,5,13)(3,22,24,8,9,25)(6,34,21,15,16,43)(7,39,40,19,20,49)(11,18,17,23,28,36)(14,37,46,32,33,47)(26,38,41,44,27,30)(29,42,31,45,35,48)(51,55,56,52,53,57)(54,63,64,58,59,73)(60,76,77,66,67,90)(61,75,71,69,65,62)(68,95,96,79,80,113)(70,93,94,82,78,87)(72,74,92,84,85,88)(81,118,119,98,99,137)(83,116,117,101,97,110)(86,91,115,103,104,111)(89,128,126,107,108,106)(100,129,130,109,121,145)(102,140,141,122,120,134)(105,114,139,123,124,135)(112,147,143,131,132,127)(125,138,146,133,142,151)(136,152,150,148,149,144);(1,23,20,31,30,35,4,8,45,14,13,40,15,19,22,46,5,34,36,28,43,24,44,38,37,33,27,26,25,21,18,17,16,10,47,9,7,6,49,12,32,29,3,2,42,41,48,39,11)(50,51,54,60,68,81,100,107,85,59,65,73,87,106,127,144,148,124,80,97,108,111,113,134,119,139,147,121,140,118,138,149,151,145,117,77,92,93,76,91,112,133,123,103,84,71,56,55,61,70,83,102,122,101,82,69,53,57,62,72,86,105,125,131,104,67,78,88,90,110,130,146,152,142,99,120,129,132,135,137,141,96,115,128,116,95,114,136,150,143,126,94,64,75,63,74,89,109,98,79,66,58,52)Polycyclic presentation:
2,3,7,103,7|c^7=e^6,c^a=c^6e,d^a=d^102,e^a=e^6,c^b=c^2e^3,d^b=d^56,e^b=e^2847 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30282, 23);