Finite Group DatabasebetaFinite Group Databasebeta| order | 30318 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C163⋊C3)⋊D31 |
| Fitting, socle, derived, Hall | C31×C163 ⋊ C2×C3 |
| Hall | C31×C163⋊C3 ⋊ C2 |
| Hall | D5053 ⋊ C3 |
| Hall | C163⋊C3 ⋊ D31 |
| Sylow | C163 ⋊ C3×D31 |
| Sylow | C31 ⋊ D163⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C31×C163 | C2×C3 |
| LCS | G ▷ C31×C163 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 194 |
| transitive degree | μt(G) = 5053 |
A minimal faithful permutation representation — 2 generators on 194 points:
(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)(33,37,38,34,35,39)(36,45,46,40,41,55)(42,58,59,48,49,72)(43,57,53,51,47,44)(50,77,78,61,62,95)(52,75,76,64,60,69)(54,56,74,66,67,70)(63,100,101,80,81,124)(65,98,99,83,79,92)(68,73,97,85,86,93)(71,113,111,89,90,88)(82,129,130,103,104,158)(84,127,128,106,102,121)(87,96,126,108,109,122)(91,142,143,114,115,117)(94,148,140,118,119,112)(105,163,164,132,133,185)(107,161,162,135,131,155)(110,125,160,137,138,156)(116,170,171,144,145,152)(120,175,134,149,150,147)(123,180,169,153,154,141)(136,186,187,166,165,183)(139,159,182,168,157,184)(146,188,189,172,173,181)(151,191,167,176,177,179)(174,192,193,190,178,194);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(32,33,36,42,50,63,82,105,134,143,99,59,74,75,58,73,94,120,151,178,193,187,130,160,180,175,170,161,129,159,138,81,102,115,119,122,124,155,181,190,172,144,114,89,67,41,47,55,69,88,112,141,168,137,108,85,66,53,38,37,43,52,65,84,107,136,167,171,128,78,97,113,98,77,96,123,133,165,173,177,164,182,184,185,191,188,186,163,153,109,62,79,90,93,95,121,152,179,166,135,106,83,64,51,35,39,44,54,68,87,110,139,169,140,111,76,46,57,45,56,71,91,116,146,174,189,162,101,126,148,142,127,100,125,157,104,131,145,150,154,156,158,183,194,192,176,149,118,86,49,60,70,72,92,117,147,132,103,80,61,48,40,34)Polycyclic presentation:
2,3,31,163|c^a=c^30,d^a=d^162,d^b=d^104888 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30318, 21);