Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×#7547 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | D32 |
| commutator | C32 | C2×C4×C17 |
| Frattini, cosocle | C2×C32 | C22×C17 |
| socle | C2×C17 | #1423 |
| prime radical | C22×C17 | Q64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C2×C8×C17 ◁ C2×C16×C17 ◁ C2×C17×C32 ◁ G | |
60 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)(18,19,22,29,26,32,25,55)(20,33,36,70,40,73,39,91)(21,41,43,92,46,95,45,101)(23,49,47,102,52,104,51,105)(24,53,48,68,54,107,50,31)(27,56,57,82,60,81,108,35)(28,61,62,78,64,77,109,34)(30,66,65,97,67,96,103,42)(37,85,83,134,88,136,87,137)(38,89,84,124,90,138,86,72)(44,99,98,141,100,142,75,94)(58,114,113,144,115,145,120,126)(59,80,106,112,116,133,143,125)(63,76,93,117,118,130,140,129)(69,111,119,128,121,127,139,74)(71,110,122,132,123,131,135,79);(18,20,34,74,23,37,75,125,50,86,129,145,96,131,45,81,25,39,77,127,51,87,100,133,54,90,130,144,97,132,46,82,26,40,78,128,52,88,98,112,48,84,117,114,66,110,43,56,22,36,61,111,47,83,44,80,24,38,76,126,42,79,21,35)(19,27,41,71,30,58,93,89,53,106,99,134,102,119,62,70,29,57,92,122,65,113,118,124,68,116,141,136,104,121,64,73,32,60,95,123,67,115,140,138,107,143,142,137,105,139,109,91,55,108,101,135,103,120,63,72,31,59,94,85,49,69,28,33)1190 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 535);