Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C181 · Schur multiplier (H2) = C22 |
| direct product | C2×SD64×C181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C181 | D16 |
| commutator, Frattini, cosocle | C16 | C23×C181 |
| prime radical | C2×D16×C181 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C181 ◁ C2×C4×C181 ◁ C2×C8×C181 ◁ C2×C16×C181 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 215 |
| transitive degree | μt(G) = 11584 |
A minimal faithful permutation representation — 3 generators on 215 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215);(3,4,8,13)(5,9,17,24)(6,18,20,34)(7,21,19,12)(10,25,27,14)(11,28,29,15)(16,30,31,23)(22,33,26,32);(4,9)(5,14)(6,15)(7,19)(10,11)(12,30)(13,24)(16,33)(17,25)(18,22)(20,28)(21,23)(26,34)(27,29)(31,32)6878 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 992);