Finite Group DatabasebetaFinite Group Databasebeta| order | 12426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C19×C109 ⋊ C2×C3 |
| Hall | #20261879 ⋊ C2 |
| Hall | C19×D109 ⋊ C3 |
| Hall | D109 ⋊ C19⋊C3 |
| Sylow | C109 ⋊ C2×C19⋊C3 |
| Sylow | C19 ⋊ D109⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C19×C109 | C2×C3 |
| LCS | G ▷ C19×C109 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 2071 |
A minimal faithful permutation representation — 2 generators on 128 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,92)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,81,79,58,59,57)(51,96,78,72,73,98)(53,94,95,74,71,89)(56,65,93,76,77,90)(60,101,102,82,83,85)(63,105,99,86,87,80)(75,108,100,91,97,109)(84,106,104,88,103,107)(111,112,114)(113,116,117)(115,119,118)(120,124,122)(121,125,123)(126,128,127);(1,2,5,11,19,32,51,50,71,83,87,90,92,109,99,79,45,15,26,14,25,40,60,84,86,55,18,29,39,41,61,85,104,95,47,66,81,67,46,65,91,74,52,33,20,4,8,13,23,37,56,78,73,97,103,106,108,96,98,76,54,35,22,7,6,12,21,34,53,75,77,31,48,59,62,64,89,107,102,68,28,43,44,27,42,63,88,82,58,36,10,16,24,38,57,80,100,70,93,105,101,94,69,72,49,30,17,9,3)(110,111,113,117,122,127,124,112,115,120,125,114,118,123,116,121,126,128,119)Polycyclic presentation:
2,3,19,109|d^a=d^108,c^b=c^7,d^b=d^45360 conjugacy classes — view the full character table.
| maximal subgroup of | #38733154, #27159092⋊C2, #38733160, … |
| maximal normal subgroup of | #38733154, #27159092⋊C2, #38733160, … |
| maximal quotient of | #38733142, C2×#27159092, #49152100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12426, 16);