Finite Group DatabasebetaFinite Group Databasebeta| order | 31434 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C31⋊C3)⋊D169 |
| Fitting, derived, Hall | C31×C169 ⋊ C2×C3 |
| Hall | C31⋊C3×C169 ⋊ C2 |
| Hall | D5239 ⋊ C3 |
| Sylow | C169 ⋊ D31⋊C3 |
| Hall | C31⋊C3 ⋊ D169 |
| Sylow | C31 ⋊ C3×D169 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C31×C169 | C2×C3 |
| Frattini | C13 | (C31⋊C3)⋊D13 |
| socle | C13×C31 | C3×D13 |
| LCS | G ▷ C31×C169 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 200 |
| transitive degree | μt(G) = 5239 |
A minimal faithful permutation representation — 2 generators on 200 points:
(2,6,7,3,4,8)(5,14,15,9,10,22)(11,24,25,17,18,30)(12,23,19,20,16,13)(21,31,28,27,26,29)(33,35)(34,38)(36,41)(37,44)(39,47)(40,46)(42,51)(43,54)(45,56)(48,60)(49,59)(50,58)(52,65)(53,68)(55,70)(57,72)(61,77)(62,76)(63,75)(64,74)(66,83)(67,86)(69,88)(71,90)(73,92)(78,98)(79,97)(80,96)(81,95)(82,94)(84,105)(85,108)(87,110)(89,112)(91,114)(93,116)(99,123)(100,122)(101,121)(102,120)(103,119)(104,118)(106,107)(109,131)(111,133)(113,135)(115,137)(117,139)(124,140)(125,145)(126,144)(127,143)(128,142)(129,141)(130,151)(132,153)(134,155)(136,157)(138,159)(146,160)(147,164)(148,163)(149,162)(150,161)(152,169)(154,171)(156,173)(158,175)(165,176)(166,179)(167,178)(168,177)(170,183)(172,185)(174,187)(180,188)(181,190)(182,189)(184,193)(186,195)(191,196)(192,197)(194,199)(198,200);(1,2,5,11,19,7,6,12,21,28,15,23,14,18,26,25,30,31,24,10,16,22,29,27,20,4,8,13,17,9,3)(32,46,75,120,163,190,200,192,167,127,81,50,35,38,59,96,144,179,196,194,182,149,103,64,41,44,47,76,121,164,188,186,184,168,128,82,51,54,56,60,97,145,176,174,172,170,150,104,65,68,70,72,77,122,160,158,156,154,152,129,83,86,88,90,92,98,140,138,136,134,132,130,105,108,110,112,114,116,123,117,115,113,111,109,107,106,131,133,135,137,139,99,93,91,89,87,85,84,151,153,155,157,159,124,78,73,71,69,67,66,141,169,171,173,175,146,100,61,57,55,53,52,118,161,183,185,187,165,125,79,48,45,43,42,94,142,177,193,195,180,147,101,62,39,37,36,74,119,162,189,199,191,166,126,80,49,34,33,58,95,143,178,197,198,181,148,102,63,40)Polycyclic presentation:
2,3,13,31,13|c^13=e^12,c^a=c^12e,d^a=d^30,e^a=e^12,d^b=d^251103 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31434, 21);