Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C71 · Schur multiplier (H2) = C22 |
| direct product | C71×#7470 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C71 | D4 |
| commutator | C4 | C4×C16×C71 |
| Frattini, cosocle | C22×C16 | C22×C71 |
| socle | C22×C71 | C22⋊C16 |
| prime radical | C23×C71 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C71 ◁ C22×C16×C71 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 9088 |
A minimal faithful permutation representation — 2 generators on 139 points:
(5,6,9,15,10,16,28,41,11,17,29,42,31,44,56,64,12,18,30,43,32,45,57,65,8,14,25,38,26,39,53,62)(7,19,21,46,22,47,50,66,20,36,48,40,49,60,67,63,23,37,27,54,51,61,55,68,24,13,33,34,52,35,58,59)(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);(1,2,3,4)(5,7)(6,13)(8,24)(9,27)(10,22)(11,20)(12,23)(14,37)(15,40)(16,35)(17,19)(18,36)(21,30)(25,48)(26,52)(28,55)(29,33)(31,49)(32,51)(34,43)(38,46)(39,61)(41,63)(42,54)(44,47)(45,60)(50,57)(53,67)(56,58)(59,65)(62,66)(64,68)7952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 458);