Finite Group DatabasebetaFinite Group Databasebeta| order | 31434 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C169⋊C3)⋊D31 |
| Fitting, derived, Hall | C31×C169 ⋊ C2×C3 |
| Hall | C31×C169⋊C3 ⋊ C2 |
| Hall | D5239 ⋊ C3 |
| Hall | C169⋊C3 ⋊ D31 |
| Sylow | C169 ⋊ C3×D31 |
| Sylow | C31 ⋊ D169⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C31×C169 | C2×C3 |
| Frattini | C13 | (C13⋊C3)⋊D31 |
| socle | C13×C31 | D13⋊C3 |
| 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:
(2,3)(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)(33,42,44,35,36,45)(34,54,56,39,40,58)(37,71,73,47,48,117)(38,83,84,51,52,137)(41,98,97,59,60,152)(43,90,101,55,63,114)(46,112,92,68,69,147)(49,128,129,76,77,186)(50,134,135,80,81,192)(53,130,143,87,88,189)(57,102,65,93,94,105)(61,122,107,82,74,66)(62,133,140,79,85,96)(64,144,157,99,103,131)(67,89,139,110,104,86)(70,171,151,115,116,191)(72,146,160,91,120,159)(75,183,148,125,126,158)(78,100,190,106,132,142)(95,170,123,113,149,179)(108,118,188,161,162,174)(109,138,127,165,166,185)(111,153,156,167,168,196)(119,193,172,164,175,150)(121,178,184,154,176,141)(124,169,195,180,181,155)(136,173,145,182,194,177)(163,187,200,198,199,197);(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)(32,55,154,196,171,187,81,182,173,143,102,72,35,39,99,145,144,128,193,127,124,117,114,63,47,51,59,89,148,88,157,135,75,74,179,69,76,80,87,54,92,188,142,137,141,48,176,181,106,62,61,93,56,129,197,86,85,162,164,163,183,65,42,113,140,152,153,155,52,111,109,108,170,160,44,68,172,191,98,100,166,70,67,66,146,120,107,110,115,138,132,60,151,150,46,45,159,149,161,165,167,83,195,168,97,96,95,36,105,126,198,119,118,133,139,200,186,58,57,82,79,78,169,178,71,184,84,190,174,147,40,53,50,49,112,123,122,125,192,131,130,158,104,41,38,37,90,101,73,180,185,175,77,103,177,64,34,33,91,94,189,194,136,134,199,116,156,121,43)Polycyclic presentation:
2,3,13,31,13|c^13=e^12,c^a=c^12e,d^a=d^30,e^a=e^12,c^b=c^3e^2,e^b=e^3919 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31434, 22);