Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C137 · Schur multiplier (H2) = C23 |
| direct product | C137×#13668 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C137 |
| prime radical | #3414×C137 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ C22×C8×C137 ◁ C2×M32×C137 ◁ G | |
284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 201 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 201 points:
(1,2,7,12)(3,13,15,42)(4,18,20,48)(5,24,27,58)(6,28,26,11)(8,30,31,14)(9,34,36,22)(10,38,40,21)(16,46,45,63)(17,47,29,43)(19,39,51,64)(23,56,54,50)(25,35,59,62)(32,57,60,53)(33,44,41,52)(37,55,49,61)(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);(1,3)(2,8)(4,19)(5,25)(6,29)(7,15)(9,35)(10,39)(11,41)(12,31)(13,18)(14,22)(16,21)(17,26)(20,51)(23,57)(24,32)(27,59)(28,33)(30,34)(36,62)(37,63)(38,45)(40,64)(42,48)(43,56)(44,61)(46,49)(47,50)(52,55)(53,54)(58,60);(4,20)(9,36)(13,42)(14,30)(18,48)(19,51)(22,34)(23,54)(35,62)(37,49)(43,47)(44,52)(46,63)(50,56)(53,57)(55,61)4247 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 6656);