Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#15691069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C11 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C11 |
| prime radical | C53×C11 | C52 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C11 ◁ C53×C11 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 261 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 2 generators on 261 points:
(1,2,5,13,31,12,22,39,33,64,58,7,20,46,76,4,8,16,34,60,30,21,15,38,70)(3,9,23,47,59,29,19,41,63,90,87,25,37,67,94,11,6,17,32,61,57,45,14,35,65)(10,26,50,80,88,56,18,42,62,91,107,52,40,66,95,28,24,36,68,89,86,44,48,77,93)(27,53,72,96,108,85,49,71,97,110,119,75,69,98,113,55,51,78,104,109,106,43,73,92,111)(54,83,100,116,120,105,81,99,117,122,125,74,103,112,124,84,82,102,114,121,118,79,101,115,123)(126,127,130,138,156,129,132,140,158,185,137,144,162,189,214,155,170,194,218,234,183,203,224,238,246)(128,133,145,171,204,136,131,141,163,195,154,143,139,159,190,153,169,161,157,186,135,147,173,188,184)(134,148,164,197,219,151,167,160,192,215,181,146,174,196,213,150,142,165,191,220,152,168,172,187,216)(149,175,205,222,240,179,166,198,225,236,178,201,193,221,239,180,176,206,217,235,182,202,199,226,233)(177,200,223,237,245,209,229,232,244,250,210,208,228,242,248,211,230,231,243,249,212,207,227,241,247)(251,252,253,254,255,256,257,258,259,260,261);(1,3,10,27,54,12,29,56,85,105,58,87,107,119,125,4,11,28,55,84,30,57,86,106,118)(2,6,18,43,74,22,45,52,53,82,7,9,24,49,79,8,19,44,75,83,21,25,26,51,81)(5,14,36,69,99,39,23,48,78,103,20,41,50,73,102,16,37,42,72,101,15,17,40,71,100)(13,32,62,92,112,33,35,66,96,114,46,47,68,97,115,34,63,77,98,116,38,67,80,104,117)(31,59,88,108,120,64,90,91,110,122,76,94,95,113,124,60,61,89,109,121,70,65,93,111,123)(126,128,134,149,177)(127,131,142,166,200)(129,136,151,179,209)(130,139,160,193,223)(132,143,168,201,229)(133,146,175,207,203)(135,152,182,212,183)(137,154,181,178,210)(138,157,187,217,237)(140,161,174,206,232)(141,164,198,227,224)(144,169,148,176,208)(145,172,205,231,194)(147,167,202,230,170)(150,180,211,155,153)(156,184,213,233,245)(158,188,197,226,244)(159,191,221,241,238)(162,173,165,199,228)(163,196,225,243,218)(171,192,222,242,189)(185,204,220,240,250)(186,215,235,247,246)(190,219,239,249,234)(195,216,236,248,214)1639 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 12);