Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33340 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C103 | C22×D4 |
| commutator | C4 | C23×C8×C103 |
| Frattini, cosocle | C2×C8 | C24×C103 |
| socle | C2×C103 | C22×C22⋊C8 |
| prime radical | #585×C103 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C103 ◁ C2×#131×C103 ◁ G | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2,6,12,7,13,27,40,8,14,28,41,5,11,25,38)(3,15,18,43,17,32,47,39,19,37,26,53,24,9,29,31)(4,10,22,35,23,36,52,61,20,33,48,57,21,34,50,59)(16,42,46,62,45,55,63,60,44,58,51,64,49,30,54,56)(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)(2,9)(4,16)(5,24)(6,26)(7,17)(8,19)(10,30)(11,37)(12,39)(13,15)(14,32)(18,28)(20,44)(21,49)(22,51)(23,45)(25,47)(27,29)(31,41)(33,55)(34,58)(35,60)(36,42)(38,43)(40,53)(46,48)(50,63)(52,54)(56,57)(59,62)(61,64);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,20)(9,30)(11,34)(12,35)(13,36)(14,33)(15,42)(17,45)(18,46)(19,44)(24,49)(25,50)(26,51)(27,52)(28,48)(29,54)(31,56)(32,55)(37,58)(38,59)(39,60)(40,61)(41,57)(43,62)(47,63)(53,64);(4,20)(10,33)(16,44)(21,23)(22,48)(30,55)(34,36)(35,57)(42,58)(45,49)(46,51)(50,52)(54,63)(56,60)(59,61)(62,64)9064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26328);