Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#13701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23×C17 |
| prime radical | #41×C17 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C17×#131 ◁ C17×#504 ◁ G | |
180 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 145 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,26,33)(20,34,41,77)(21,42,46,69)(22,47,51,106)(23,32,25,30)(24,54,55,70)(27,56,61,37)(28,52,64,44)(29,65,68,35)(31,53,71,48)(36,82,86,126)(38,76,40,74)(39,92,93,127)(43,97,100,140)(45,101,102,63)(49,105,50,104)(57,111,113,135)(58,88,60,89)(59,116,117,130)(62,118,119,131)(66,80,67,81)(72,90,125,84)(73,110,115,78)(75,91,128,87)(79,95,120,121)(83,94,109,114)(85,136,137,124)(96,108,122,98)(99,103,107,139)(112,129,132,141)(123,142,143,145)(133,138,144,134);(18,20,35,78,24,39,49,88,23,38,80,116,53,91,51,56,26,41,65,110,55,93,50,89,25,40,81,130,48,87,22,37)(19,27,47,75,31,59,66,74,30,58,104,127,70,115,68,77,33,61,106,128,71,117,67,76,32,60,105,92,54,73,29,34)(21,36,79,129,45,85,107,111,52,90,118,142,108,138,100,109,46,86,120,141,102,137,99,135,44,84,131,145,98,134,43,83)(28,57,103,124,63,112,121,126,69,114,140,144,122,143,119,125,64,113,139,136,101,132,95,82,42,94,97,133,96,123,62,72);(18,21)(19,28)(20,36)(22,43)(23,52)(24,45)(25,44)(26,46)(27,57)(29,62)(30,69)(31,63)(32,42)(33,64)(34,72)(35,79)(37,83)(38,90)(39,85)(40,84)(41,86)(47,103)(48,98)(49,107)(50,99)(51,100)(53,108)(54,96)(55,102)(56,109)(58,114)(59,112)(60,94)(61,113)(65,120)(66,121)(67,95)(68,119)(70,122)(71,101)(73,123)(74,126)(75,124)(76,82)(77,125)(78,129)(80,118)(81,131)(87,134)(88,111)(89,135)(91,138)(92,133)(93,137)(97,105)(104,140)(106,139)(110,141)(115,143)(116,142)(117,132)(127,144)(128,136)(130,145)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6689);