Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C181 · Schur multiplier (H2) = C2×C4 |
| direct product | #1838×C181 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C181 | C2×D4 |
| commutator | C2×C4 | C22×C4×C181 |
| Frattini, cosocle | C22×C4 | C23×C181 |
| socle | C22×C181 | C2×C4⋊C4 |
| prime radical | C23×C181 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C181 ◁ C2×C42×C181 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 205 |
| transitive degree | μt(G) = 11584 |
A minimal faithful permutation representation — 3 generators on 205 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,23)(11,20,21,24,22,17,12,16)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205);(2,5)(6,8)(9,11,13,12,15,22,14,21)(10,16,18,20,19,24,23,17);(1,3,4,7)(2,6,5,8)(9,12,13,22,15,21,14,11)(10,17,18,16,19,20,23,24)5792 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 570);