Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#15691111 |
| 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:
(1,2,5,12,26)(3,7,14,28,48)(4,8,15,29,47)(6,16,35,59,46)(9,19,32,51,72)(10,20,33,52,71)(11,21,34,27,49)(13,30,53,25,17)(18,36,60,83,70)(22,39,56,75,94)(23,40,57,76,93)(24,41,58,50,73)(31,54,77,45,37)(38,61,84,104,92)(42,64,80,97,112)(43,65,81,98,111)(44,66,82,74,95)(55,78,99,69,62)(63,85,105,119,110)(67,88,102,115,122)(68,89,103,96,113)(79,100,116,91,86)(87,106,120,125,121)(90,108,118,114,123)(101,117,124,109,107)(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);(2,6,17,21,8)(5,13,15,16,34)(7,18,37,41,20)(12,27,35,29,30)(14,31,33,36,58)(19,38,62,66,40)(26,47,49,53,59)(28,50,60,52,54)(32,55,57,61,82)(39,63,86,89,65)(48,71,73,77,83)(51,74,84,76,78)(56,79,81,85,103)(64,87,107,108,88)(72,93,95,99,104)(75,96,105,98,100)(80,101,102,106,118)(94,111,113,116,119)(97,114,120,115,117)(112,122,123,124,125)(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);(1,3,9,22,42,11,24,44,68,90,46,70,92,110,121,4,10,23,43,67,25,45,69,91,109)(2,7,19,39,64,21,41,66,89,108,6,18,38,63,87,8,20,40,65,88,17,37,62,86,107)(5,14,32,56,80,34,58,82,103,118,16,36,61,85,106,15,33,57,81,102,13,31,55,79,101)(12,28,51,75,97,27,50,74,96,114,35,60,84,105,120,29,52,76,98,115,30,54,78,100,117)(26,48,72,94,112,49,73,95,113,123,59,83,104,119,125,47,71,93,111,122,53,77,99,116,124)(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, 54);