Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C181 · Schur multiplier (H2) = C22 |
| direct product | #2171×C181 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C181 | C2×D4 |
| commutator | C4 | C22×C8×C181 |
| Frattini, cosocle | C2×C8 | C23×C181 |
| socle | C2×C181 | C8×D4 |
| prime radical | #135×C181 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C181 ◁ C2×C16×C181 ◁ G | |
144 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 5792 |
A minimal faithful permutation representation — 3 generators on 213 points:
(1,2,6,12,7,11,24,30,8,13,25,15,5,21,23,31)(3,14,17,19,16,9,26,20,10,27,29,18,22,28,32,4)(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)(2,9)(4,15)(5,22)(6,17)(7,16)(8,10)(11,27)(12,20)(13,28)(14,21)(18,30)(19,31)(23,32)(24,26)(25,29);(1,4,6,14,7,19,24,9,8,20,25,27,5,18,23,28)(2,10,12,29,11,22,30,32,13,3,15,17,21,16,31,26)7964 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 903);