Finite Group DatabasebetaFinite Group Databasebeta| order | 31650 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, socle, derived, Hall | C52×C211 ⋊ C2×C3 |
| Hall | C52⋊(C211⋊C3) ⋊ C2 |
| Hall | C5⋊D1055 ⋊ C3 |
| Sylow | C211 ⋊ (C5⋊D5)⋊C3 |
| Sylow | C52 ⋊ D211⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C211 | C2×C3 |
| LCS | G ▷ C52×C211 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 226 |
| transitive degree | μt(G) = 3165 |
A minimal faithful permutation representation — 2 generators on 226 points:
(1,2,4)(3,13,9,7,6,10)(5,15,14,11,12,8)(17,21,22,18,19,23)(20,29,30,24,25,39)(26,42,43,32,33,56)(27,41,37,35,31,28)(34,61,62,45,46,79)(36,59,60,48,44,53)(38,40,58,50,51,54)(47,84,85,64,65,108)(49,82,83,67,63,76)(52,57,81,69,70,77)(55,97,95,73,74,72)(66,113,114,87,88,143)(68,111,112,90,86,105)(71,80,110,92,93,106)(75,126,127,98,99,101)(78,132,124,102,103,96)(89,148,149,116,117,180)(91,146,147,119,115,140)(94,109,145,121,122,141)(100,157,158,128,129,136)(104,164,162,133,134,131)(107,169,155,137,138,125)(118,150,177,120,151,183)(123,144,182,152,153,178)(130,187,188,159,160,173)(135,191,192,165,166,168)(139,197,189,170,171,163)(142,202,185,174,175,156)(154,181,206,161,184,210)(167,213,214,193,194,201)(172,217,215,198,199,196)(176,219,211,203,204,190)(179,223,195,207,208,186)(200,224,225,212,209,218)(205,226,222,220,221,216);(1,3,8,14,7)(2,5,6,13,11)(4,9,15,12,10)(16,17,20,26,34,47,66,89,118,119,90,67,48,35,19,23,28,38,52,71,94,123,154,159,128,98,73,51,25,31,39,53,72,96,125,156,186,193,165,133,102,70,33,44,54,56,76,101,131,163,190,212,198,170,137,93,46,63,74,77,79,105,136,168,196,216,220,203,174,122,65,86,99,103,106,108,140,173,201,218,226,224,207,153,88,115,129,134,138,141,143,177,206,223,219,217,213,184,117,150,160,166,171,175,178,180,149,182,202,197,191,187,151,148,181,194,199,204,208,210,183,114,145,169,164,157,146,113,144,179,209,221,225,214,188,147,85,110,132,126,111,84,109,142,176,205,222,215,192,158,112,62,81,97,82,61,80,107,139,172,200,211,189,162,127,83,43,58,59,42,57,78,104,135,167,195,185,155,124,95,60,30,41,29,40,55,75,100,130,161,152,121,92,69,50,37,22,21,27,36,49,68,91,120,116,87,64,45,32,24,18)Polycyclic presentation:
2,3,5^2,211|c^a=c^4,d^a=d^4,e^a=e^210,c^b=cd^3,d^b=c^4d^3,e^b=e^14885 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31650, 36);