Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691110 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C7 | C53 |
| commutator, Frattini, cosocle | C52 | C53×C7 |
| prime radical | C53×C7 | C52 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C7 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 257 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 3 generators on 257 points:
(1,2,6,16,37)(3,10,26,55,94)(4,8,18,39,69)(5,9,19,40,70)(7,20,47,85,63)(11,29,59,78,114)(12,28,56,95,120)(13,23,44,74,68)(14,24,45,75,104)(15,25,46,76,105)(17,41,77,67,93)(21,50,89,106,101)(22,49,86,118,100)(27,57,72,108,99)(30,61,88,119,123)(31,60,96,115,125)(32,58,42,80,116)(33,51,82,111,103)(34,52,83,38,71)(35,53,81,112,121)(36,54,84,110,122)(43,79,113,62,98)(48,87,102,65,91)(64,97,90,109,124)(66,92,117,73,107)(126,127,130,137,151)(128,131,138,152,172)(129,133,140,154,173)(132,144,166,191,171)(134,141,155,174,196)(135,143,157,176,197)(136,146,159,153,177)(139,158,183,150,145)(142,163,190,214,195)(147,160,178,198,218)(148,162,180,200,219)(149,165,182,175,201)(156,181,207,170,164)(161,187,213,233,217)(167,184,202,220,236)(168,186,204,222,237)(169,189,206,199,223)(179,205,228,194,188)(185,210,232,245,235)(192,209,225,239,247)(193,212,227,221,240)(203,226,243,216,211)(208,229,244,250,246)(215,231,242,238,248)(224,241,249,234,230)(251,252,253,254,255,256,257);(1,3,11,30,62,15,34,65,4,12,31,63,99,35,66,5,13,32,64,100,36,67,101,14,33)(2,7,21,8,22,25,23,29,53,51,60,93,91,97,98,9,10,27,24,28,54,52,58,61,92)(6,17,42,81,56,46,26,50,90,117,96,83,59,45,49,19,20,48,88,82,84,44,57,18,43)(16,38,72,109,111,76,47,80,75,79,115,74,89,119,95,40,41,78,39,73,110,55,87,112,86)(37,68,102,104,107,105,77,108,123,118,125,94,116,69,103,70,71,106,121,113,122,85,114,124,120)(126,128,134,147,167,129,135,148,168,192,136,149,169,193,215,150,170,194,216,234,171,195,217,235,246)(127,131,141,160,184,133,143,162,186,209,146,165,189,212,231,145,164,188,211,230,132,142,161,185,208)(130,138,155,178,202,140,157,180,204,225,159,182,206,227,242,139,156,179,203,224,144,163,187,210,229)(137,152,174,198,220,154,176,200,222,239,153,175,199,221,238,158,181,205,226,241,166,190,213,232,244)(151,172,196,218,236,173,197,219,237,247,177,201,223,240,248,183,207,228,243,249,191,214,233,245,250);(1,4,13,33,65,5,14,34,66,101,15,35,67,62,99,36,30,63,100,11,31,64,3,12,32)(2,8,23,51,91,9,24,52,92,21,25,53,93,98,27,54,61,7,22,29,60,97,10,28,58)(6,18,44,82,48,19,45,83,117,50,46,81,17,43,57,84,88,20,49,59,96,90,26,56,42)(16,39,74,111,87,40,75,38,73,89,76,112,41,79,72,110,119,47,86,78,115,109,55,95,80)(37,69,68,103,102,70,104,71,107,106,105,121,77,113,108,122,123,85,118,114,125,124,94,120,116)(127,132,145,146,133)(130,139,140,144,159)(131,142,164,165,143)(137,153,166,154,158)(138,156,157,163,182)(141,161,188,189,162)(151,173,177,183,191)(152,175,190,176,181)(155,179,180,187,206)(160,185,211,212,186)(172,197,201,207,214)(174,199,213,200,205)(178,203,204,210,227)(184,208,230,231,209)(196,219,223,228,233)(198,221,232,222,226)(202,224,225,229,242)(218,237,240,243,245)(220,238,244,239,241)(236,247,248,249,250)1715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 53);