Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C25 |
| direct product | C2×#2222×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C131 |
| prime radical | C2×#41×C131 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C2×#131×C131 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 197 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 4 generators on 197 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197);(3,4,10,16)(5,17,23,45)(6,24,27,41)(7,28,30,39)(8,15,9,14)(11,32,36,20)(12,31,38,26)(13,29,40,18)(19,48,51,65)(21,44,22,43)(25,56,57,37)(33,60,61,66)(34,52,35,46)(42,53,64,50)(47,55,58,63)(49,54,59,62);(3,5,18,46,9,22,30,32,10,23,29,52,8,21,7,20)(4,11,28,44,15,35,40,45,16,36,39,43,14,34,13,17)(6,19,47,66,26,50,57,59,27,51,58,60,31,53,25,49)(12,33,55,48,24,54,56,64,38,61,63,65,41,62,37,42);(3,6)(4,12)(5,19)(7,25)(8,31)(9,26)(10,27)(11,33)(13,37)(14,41)(15,24)(16,38)(17,42)(18,47)(20,49)(21,53)(22,50)(23,51)(28,55)(29,58)(30,57)(32,59)(34,62)(35,54)(36,61)(39,63)(40,56)(43,65)(44,48)(45,64)(46,66)(52,60)6812 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26603);