Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C2×#41×C103 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C22×C4×C103 ◁ C2×#131×C103 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2,7,12)(3,8,14,30)(4,17,18,52)(5,21,26,57)(6,27,24,11)(9,33,34,19)(10,36,39,25)(13,41,42,51)(15,46,22,56)(16,49,45,32)(20,38,28,53)(23,35,55,40)(29,54,60,43)(31,59,37,48)(44,62,50,64)(47,61,58,63)(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)(2,8)(4,13)(5,22)(6,16)(7,14)(9,29)(10,37)(11,32)(12,30)(15,26)(17,51)(18,42)(19,54)(20,44)(21,56)(23,58)(24,45)(25,59)(27,49)(28,50)(31,39)(33,43)(34,60)(35,61)(36,48)(38,64)(40,63)(41,52)(46,57)(47,55)(53,62);(1,4,6,28,7,18,24,20)(2,9,11,40,12,34,27,35)(3,13,16,50,14,42,45,44)(5,23,25,33,26,55,36,19)(8,29,32,63,30,60,49,61)(10,38,21,52,39,53,57,17)(15,47,48,54,22,58,59,43)(31,62,46,51,37,64,56,41);(3,14)(8,30)(13,42)(15,22)(16,45)(29,60)(31,37)(32,49)(41,51)(43,54)(44,50)(46,56)(47,58)(48,59)(61,63)(62,64)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26646);