Finite Group DatabasebetaFinite Group Databasebeta| order | 34688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C271 · Schur multiplier (H2) = C22 |
| direct product | #1382×C271 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C271 | D8 |
| commutator | C8 | C42×C271 |
| Frattini, cosocle | C22×C8 | C22×C271 |
| socle | C22×C271 | #102 |
| prime radical | C23×C271 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C271 ◁ C22×C4×C271 ◁ C22×C8×C271 ◁ G | |
132 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 307 |
| transitive degree | μt(G) = 17344 |
A minimal faithful permutation representation — 2 generators on 307 points:
(1,2,3,4)(5,6)(7,16)(8,21)(9,24)(10,26)(11,15)(12,17)(13,23)(14,22)(18,28)(19,36)(20,32)(25,29)(27,35)(30,34)(31,33)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307);(5,7,9,18,22,33,23,34,11,20,8,17,10,19,25,35)(6,12,13,16,26,30,24,36,15,28,29,32,14,27,21,31)11924 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34688, 114);