Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#25448 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C103×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C22×C4×C103 ◁ G | |
860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2,8,12,7,14,6,13)(3,15,29,34,19,33,18,9)(4,20,23,10,25,40,24,37)(5,26,30,39,28,38,27,11)(16,35,44,41,21,31,45,49)(17,46,56,58,48,32,47,42)(22,52,53,50,55,36,54,60)(43,62,63,59,51,61,64,57)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,3,7,19)(2,9)(4,21)(5,17,28,48)(6,18,8,29)(10,35,37,31)(11,32)(12,15)(13,33)(14,34)(16,25)(20,49,40,41)(22,51)(23,45)(24,44)(26,42)(27,47,30,56)(36,59,52,57)(38,58)(39,46)(43,55)(50,62,60,61)(53,64)(54,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,25)(8,24)(9,31)(11,36)(12,20)(13,40)(14,37)(15,41)(17,43)(18,44)(19,21)(26,50)(27,53)(28,55)(29,45)(30,54)(32,57)(33,49)(34,35)(38,60)(39,52)(42,61)(46,59)(47,63)(48,51)(56,64)(58,62);(1,5,8,30,7,28,6,27)(2,11,13,38,14,39,12,26)(3,17,29,56,19,48,18,47)(4,22,24,54,25,55,23,53)(9,32,33,58,34,46,15,42)(10,36,40,60,37,52,20,50)(16,43,45,64,21,51,44,63)(31,57,49,62,35,59,41,61)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 18436);