Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C71 · Schur multiplier (H2) = C22 |
| direct product | C71×#7469 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C71 | C22⋊C4 |
| commutator | C22 | C4×C16×C71 |
| Frattini, cosocle | C22×C16 | C22×C71 |
| socle | C2×C71 | #1374 |
| prime radical | C23×C71 | C2×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C71 ◁ C22×C16×C71 ◁ G | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 2 generators on 135 points:
(1,2,4,9,5,11,18,29,6,12,19,30,21,33,42,53,7,10,20,28,22,31,43,51,23,32,44,52,45,54,60,63)(3,13,15,34,16,35,39,55,17,36,40,56,41,57,59,64,14,8,24,25,37,26,46,48,38,27,47,49,58,50,61,62)(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);(1,3)(2,8,10,13)(4,15)(5,16)(6,17)(7,14)(9,25,28,34)(11,26,31,35)(12,27,32,36)(18,39)(19,40)(20,24)(21,41)(22,37)(23,38)(29,48,51,55)(30,49,52,56)(33,50,54,57)(42,59)(43,46)(44,47)(45,58)(53,62,63,64)(60,61)6248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 457);