Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C137 · Schur multiplier (H2) = C23 |
| direct product | C137×#11588 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C137 |
| Frattini, cosocle | C4×C8 | C23×C137 |
| socle | C22×C137 | C2×C22⋊C8 |
| prime radical | #532×C137 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C2×C4×C8×C137 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 177 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 3 generators on 177 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,17,14,18,25,32,13,16,24,29,26,30,37,39)(11,19,21,31,22,33,35,40,20,15,23,27,34,28,36,38)(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);(2,5)(3,8)(6,7)(9,11)(10,15)(12,23)(13,20)(14,22)(16,19)(17,31)(18,28)(21,24)(25,36)(26,34)(27,29)(30,33)(32,40)(35,37)(38,39);(10,16)(11,20)(17,29)(18,30)(21,23)(22,34)(32,39)(35,36)8768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 4576);