Finite Group DatabasebetaFinite Group Databasebeta| order | 23322 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C23⋊(D169⋊C3) |
| unambiguous semidirect product | (C169⋊C3)⋊D23 |
| unambiguous semidirect product | D3887⋊C3 |
| Fitting, derived, Hall | C23×C169 ⋊ C2×C3 |
| Hall | C23×C169⋊C3 ⋊ C2 |
| Hall | D3887 ⋊ C3 |
| Hall | C169⋊C3 ⋊ D23 |
| Sylow | C169 ⋊ C3×D23 |
| Sylow | C23 ⋊ D169⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C23×C169 | C2×C3 |
| Frattini | C13 | C23⋊(D13⋊C3) |
| socle | C13×C23 | D13⋊C3 |
| LCS | G ▷ C23×C169 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 192 |
| transitive degree | μt(G) = 3887 |
A minimal faithful permutation representation — 2 generators on 192 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,34,36,27,28,37)(26,46,48,31,32,50)(29,63,65,39,40,109)(30,75,76,43,44,129)(33,89,91,51,52,143)(35,83,98,47,55,107)(38,105,81,60,61,138)(41,119,120,68,69,155)(42,124,125,72,73,180)(45,134,135,79,80,176)(49,99,57,85,86,102)(53,113,74,93,66,58)(54,131,101,96,77,88)(56,144,154,90,100,141)(59,82,130,71,104,78)(62,122,146,70,108,178)(64,106,164,84,111,158)(67,172,128,116,117,156)(87,165,114,140,123,169)(92,189,103,147,137,157)(94,177,153,149,121,159)(95,182,148,152,126,185)(97,188,175,133,136,191)(110,181,167,161,162,142)(112,186,170,145,166,115)(118,168,187,173,174,132)(127,160,139,183,151,190)(150,192,163,171,179,184);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,47,145,188,189,184,125,162,110,80,105,64,27,31,90,142,141,155,127,126,168,63,83,98,39,43,51,88,87,135,100,73,114,113,116,86,68,72,79,50,49,121,108,75,186,65,170,173,62,59,58,138,32,69,192,131,130,153,151,150,165,38,37,156,104,89,191,187,76,97,95,94,172,111,28,102,190,147,143,146,148,103,54,53,158,164,93,96,157,185,178,91,92,139,57,34,106,117,149,152,133,129,132,175,52,82,128,36,60,123,171,160,159,78,77,179,119,46,81,74,71,70,118,115,109,166,44,122,177,85,48,45,42,41,99,67,66,169,124,144,176,140,101,33,30,29,107,55,40,174,182,183,120,154,167,56,26,25,84,61,134,161,181,180,163,137,136,112,35)Polycyclic presentation:
2,3,13,23,13|c^13=e^12,c^a=c^12e,d^a=d^22,e^a=e^12,c^b=c^9e^12,e^b=e^9683 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23322, 4);