Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C181 · Schur multiplier (H2) = C2 |
| direct product | #1321×C181 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C181 | #99 |
| commutator | C2×C4 | C2×C8×C181 |
| Frattini, cosocle | C2×M16 | C22×C181 |
| socle | C2×C181 | #323 |
| prime radical | C2×#41×C181 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C181 ◁ C2×C4×C181 ◁ C2×M16×C181 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 5792 |
A minimal faithful permutation representation — 2 generators on 213 points:
(1,2,5,10,6,9,20,26,7,11,21,28,4,16,18,27)(3,12,14,29,13,8,22,23,15,25,19,31,17,24,30,32)(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)(2,8)(4,17)(5,19)(6,13)(7,15)(9,25)(10,23)(11,24)(12,16)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32)4706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 53);