Finite Group DatabasebetaFinite Group Databasebeta| order | 21875 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C54×C7 · Schur multiplier (H2) = C55 |
| direct product | C5×C7×#10597837 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C7×C25 | C52 |
| commutator, Frattini, cosocle | C5 | C54×C7 |
| socle | C52×C7 | C53 |
| prime radical | C5×C7×H(3,𝔽5) | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C7×C25 ◁ G | |
2372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4375 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,3,4,5)(131,132,133,134,135,136,137);(6,7,10,17,31)(8,12,19,33,53)(9,13,20,34,52)(11,21,40,64,51)(14,24,37,56,77)(15,25,38,57,76)(16,26,39,32,54)(18,35,58,30,22)(23,41,65,88,75)(27,44,61,80,99)(28,45,62,81,98)(29,46,63,55,78)(36,59,82,50,42)(43,66,89,109,97)(47,69,85,102,117)(48,70,86,103,116)(49,71,87,79,100)(60,83,104,74,67)(68,90,110,124,115)(72,93,107,120,127)(73,94,108,101,118)(84,105,121,96,91)(92,111,125,130,126)(95,113,123,119,128)(106,122,129,114,112);(7,11,22,26,13)(10,18,20,21,39)(12,23,42,46,25)(17,32,40,34,35)(19,36,38,41,63)(24,43,67,71,45)(31,52,54,58,64)(33,55,65,57,59)(37,60,62,66,87)(44,68,91,94,70)(53,76,78,82,88)(56,79,89,81,83)(61,84,86,90,108)(69,92,112,113,93)(77,98,100,104,109)(80,101,110,103,105)(85,106,107,111,123)(99,116,118,121,124)(102,119,125,120,122)(117,127,128,129,130);(6,8,14,27,47,9,15,28,48,72,16,29,49,73,95,30,50,74,96,114,51,75,97,115,126)(7,12,24,44,69,13,25,45,70,93,26,46,71,94,113,22,42,67,91,112,11,23,43,68,92)(10,19,37,61,85,20,38,62,86,107,39,63,87,108,123,18,36,60,84,106,21,41,66,90,111)(17,33,56,80,102,34,57,81,103,120,32,55,79,101,119,35,59,83,105,122,40,65,89,110,125)(31,53,77,99,117,52,76,98,116,127,54,78,100,118,128,58,82,104,121,129,64,88,109,124,130)5075 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21875, 74);