Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×#15691113 |
| 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:
(1,2,6,17,39)(3,10,27,55,88)(4,8,19,40,66)(5,9,20,41,67)(7,21,47,78,37)(11,30,44,71,96)(12,28,48,79,90)(13,29,56,89,65)(14,24,45,68,92)(15,25,18,42,70)(16,26,46,69,93)(22,49,81,91,35)(23,43,72,98,33)(31,51,80,107,110)(32,50,75,97,115)(34,57,73,100,109)(36,53,76,94,111)(38,54,77,95,112)(52,74,99,108,61)(58,85,103,117,121)(59,84,101,119,123)(60,83,104,116,120)(62,82,102,118,122)(63,86,105,113,124)(64,87,106,114,125)(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);(1,3,11,31,58,16,4,12,32,59,64,37,14,34,60,5,13,33,36,62,38,15,35,61,63)(2,7,22,50,82,26,29,30,57,86,87,25,28,53,85,9,10,24,52,84,54,8,23,51,83)(6,18,43,73,101,46,27,49,76,104,106,19,44,74,102,20,21,48,80,105,77,56,45,75,103)(17,40,68,94,113,69,47,72,99,117,114,89,81,107,119,41,42,71,97,116,95,55,79,100,118)(39,65,90,108,120,93,70,92,110,122,125,88,98,115,124,67,66,91,109,121,112,78,96,111,123)(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);(1,4,14,36,63,16,37,33,61,58,64,13,35,31,59,5,15,11,32,60,38,3,12,34,62)(2,8,24,53,86,26,7,23,52,85,87,29,22,51,84,9,25,30,50,83,54,10,28,57,82)(6,19,45,76,105,46,21,43,74,103,106,56,49,80,101,20,18,44,75,104,77,27,48,73,102)(17,40,68,94,113,69,47,72,99,117,114,89,81,107,119,41,42,71,97,116,95,55,79,100,118)(39,66,92,111,124,93,78,98,108,121,125,65,91,110,123,67,70,96,115,120,112,88,90,109,122)(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, 56);