Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C181 · Schur multiplier (H2) = C23 |
| direct product | #1830×C181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C181 | #381 |
| commutator | C23 | C22×C4×C181 |
| Frattini, cosocle | C22×C4 | C23×C181 |
| prime radical | C23×C181 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C181 ◁ C22×C4×C181 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 5792 |
A minimal faithful permutation representation — 3 generators on 213 points:
(1,2,5,10,6,9,19,24)(3,11,13,25,14,7,18,21)(4,15,17,29,16,8,22,23)(12,20,28,31,27,26,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3,6,14)(2,7,9,11)(4,12,16,27)(5,18,19,13)(8,20,15,26)(10,25,24,21)(17,30,22,28)(23,32,29,31);(1,4)(2,8,9,15)(3,12)(5,17)(6,16)(7,20,11,26)(10,23,24,29)(13,28)(14,27)(18,30)(19,22)(21,31,25,32)5249 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 562);