Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C181 · Schur multiplier (H2) = C23 |
| direct product | #1828×C181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C181 | #381 |
| commutator | C23 | C22×C4×C181 |
| Frattini, cosocle | C24 | C23×C181 |
| prime radical | C22×D4×C181 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C181 ◁ C24×C181 ◁ G | |
296 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 197 |
| transitive degree | μt(G) = 2896 |
A minimal faithful permutation representation — 3 generators on 197 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)5249 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 560);