Finite Group DatabasebetaFinite Group Databasebeta| order | 32754 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C53⋊(D103⋊C3) |
| unambiguous semidirect product | (C103⋊C3)⋊D53 |
| unambiguous semidirect product | D5459⋊C3 |
| Fitting, socle, derived, Hall | C53×C103 ⋊ C2×C3 |
| Hall | C53×C103⋊C3 ⋊ C2 |
| Hall | D5459 ⋊ C3 |
| Hall | C103⋊C3 ⋊ D53 |
| Sylow | C103 ⋊ C3×D53 |
| Sylow | C53 ⋊ D103⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C53×C103 | C2×C3 |
| LCS | G ▷ C53×C103 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 156 |
| transitive degree | μt(G) = 5459 |
A minimal faithful permutation representation — 2 generators on 156 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(55,59,60,56,57,61)(58,67,68,62,63,77)(64,80,81,70,71,94)(65,79,75,73,69,66)(72,99,100,83,84,117)(74,97,98,86,82,91)(76,78,96,88,89,92)(85,122,123,102,103,141)(87,120,121,105,101,114)(90,95,119,107,108,115)(93,132,130,111,112,110)(104,133,134,113,125,149)(106,144,145,126,124,138)(109,118,143,127,128,139)(116,151,147,135,136,131)(129,142,150,137,146,155)(140,156,154,152,153,148);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,58,64,72,85,104,111,89,63,69,77,91,110,131,148,152,128,84,101,112,115,117,138,123,143,151,125,144,122,142,153,155,149,121,81,96,97,80,95,116,137,127,107,88,75,60,59,65,74,87,106,126,105,86,73,57,61,66,76,90,109,129,135,108,71,82,92,94,114,134,150,156,146,103,124,133,136,139,141,145,100,119,132,120,99,118,140,154,147,130,98,68,79,67,78,93,113,102,83,70,62,56)Polycyclic presentation:
2,3,53,103|c^a=c^52,d^a=d^102,d^b=d^56985 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32754, 12);