Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C23 |
| direct product | C137×#12925 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | #447 |
| commutator | C2×C8 | C22×C4×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| prime radical | C2×#41×C137 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×#131×C137 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2,4,3)(5,6)(7,18)(8,22)(9,16)(10,15)(11,17)(12,19)(13,20)(14,25)(21,33)(23,34)(24,28)(26,32)(27,31)(29,36)(30,35)(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);(1,3,4,2)(5,7)(6,12)(8,23)(9,20)(10,27)(11,21)(13,28)(14,31)(15,29)(16,34)(17,30)(18,22)(19,25)(24,35)(26,36)(32,33);(5,7)(6,13)(8,19)(9,27)(10,20)(11,21)(12,28)(14,30)(15,33)(16,18)(17,31)(22,34)(23,25)(24,36)(26,35)(29,32)6302 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 5913);