Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C137 · Schur multiplier (H2) = C2×C4 |
| direct product | C137×#13728 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C137 |
| prime radical | C2×D32×C137 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C42×C137 ◁ C4×C8×C137 ◁ C4×C16×C137 ◁ G | |
238 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 205 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 3 generators on 205 points:
(2,4)(5,6)(7,17)(8,23)(9,13)(10,29)(11,30)(12,19)(14,18)(15,24)(16,28)(20,37)(21,50)(22,51)(25,52)(26,56)(27,57)(31,46)(32,60)(33,61)(34,42)(35,55)(36,62)(38,41)(39,45)(40,49)(43,54)(44,53)(47,58)(48,67)(59,68)(63,65)(64,66)(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);(5,7,18,41,10,21,43,61,28,49,62,68,55,66,25,46,9,20,42,65,27,48,26,47,11,22,44,60,24,45,8,19)(6,12,23,39,15,32,53,51,30,58,56,67,57,63,34,37,13,31,52,64,35,59,36,40,16,33,54,50,29,38,14,17);(1,2,3,4)(6,13)(12,31)(14,34)(15,35)(16,30)(17,37)(23,52)(29,57)(32,59)(33,58)(36,53)(38,63)(39,64)(40,51)(50,67)(54,56)9590 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 6716);