Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C17×C32 · Schur multiplier (H2) = C2 |
| direct product | C17×#7514 |
| subgroup | G/H | |
|---|---|---|
| center | C17×C32 | D4 |
| commutator | C4 | C2×C17×C32 |
| Frattini, cosocle | C2×C32 | C22×C17 |
| socle | C2×C17 | C22⋊C32 |
| prime radical | #41×C17 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C17×C32 ◁ C2×C17×C32 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 2 generators on 145 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,29,23,30,45,64,24,31,46,65,49,68,92,113,25,32,47,66,50,69,93,114,52,71,95,116,97,118,133,141,26,33,48,67,51,70,94,115,53,72,96,117,98,119,134,142,21,28,41,60,42,61,86,108,43,62,87,109,88,110,129,139)(20,34,36,73,37,74,79,120,38,75,80,121,81,122,126,143,35,58,76,63,77,104,123,111,78,105,124,112,125,137,144,140,39,59,44,89,82,106,90,130,83,107,91,131,127,138,132,145,40,27,54,55,84,56,99,101,85,57,100,102,128,103,135,136);(18,20)(19,27)(21,40)(22,44)(23,37)(24,38)(25,35)(26,39)(28,59)(29,63)(30,56)(31,57)(32,34)(33,58)(36,48)(41,76)(42,84)(43,85)(45,90)(46,91)(47,54)(49,81)(50,77)(51,82)(52,78)(53,83)(55,67)(60,73)(61,106)(62,107)(64,111)(65,112)(66,89)(68,103)(69,74)(70,104)(71,75)(72,105)(79,94)(80,96)(86,123)(87,124)(88,128)(92,132)(93,99)(95,100)(97,125)(98,127)(101,115)(102,117)(108,120)(109,121)(110,138)(113,140)(114,130)(116,131)(118,122)(119,137)(126,134)(129,144)(133,135)(136,142)(139,143)(141,145)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 502);