Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C71 · Schur multiplier (H2) = C22 |
| direct product | C71×#7537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | D32 |
| commutator | C32 | C2×C4×C71 |
| Frattini, cosocle | C2×C32 | C22×C71 |
| prime radical | C2×D32×C71 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C2×C4×C71 ◁ C2×C8×C71 ◁ C2×C16×C71 ◁ C2×C32×C71 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 9088 |
A minimal faithful permutation representation — 2 generators on 139 points:
(1,2,3,4)(5,6)(7,12)(8,23)(9,27)(10,30)(11,17)(13,18)(14,19)(15,24)(16,28)(20,40)(21,42)(22,34)(25,58)(26,56)(29,60)(31,44)(32,43)(33,51)(35,50)(36,53)(37,46)(38,59)(39,49)(41,57)(45,61)(47,55)(48,54)(52,65)(62,68)(63,67)(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);(6,12)(7,18)(8,19)(9,24)(10,28)(13,14)(15,40)(16,42)(17,34)(20,50)(21,53)(22,46)(23,31)(25,59)(26,49)(27,32)(29,57)(30,33)(35,55)(36,54)(37,39)(38,61)(41,65)(43,44)(45,51)(47,48)(52,68)(56,63)(58,62)(60,64)(66,67)4970 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 525);