Finite Group DatabasebetaFinite Group Databasebeta| order | 34688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C271 · Schur multiplier (H2) = C25 |
| direct product | #2916×C271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C271 | C22×D4 |
| commutator | C4 | C23×C4×C271 |
| Frattini, cosocle | C2×C4 | C24×C271 |
| socle | C2×C271 | C22×C4⋊C4 |
| prime radical | C2×#41×C271 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C271 ◁ C2×#41×C271 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 303 |
| transitive degree | μt(G) = 8672 |
A minimal faithful permutation representation — 4 generators on 303 points:
(1,2,6,10,7,11,5,20)(3,12,15,22,16,26,14,8)(4,9,19,24,17,23,18,30)(13,25,29,31,27,32,28,21)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303);(1,3,5,14,7,16,6,15)(2,8,10,12,11,22,20,26)(4,13,18,28,17,27,19,29)(9,21,24,25,23,31,30,32);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,28)(15,29)(16,27)(20,30)(22,31)(26,32);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(25,32)(28,29)11924 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34688, 1648);