metabelian, soluble, monomial, A-group
Aliases: D19⋊A4, C19⋊A4⋊C2, C19⋊(C2×A4), (C2×C38)⋊2C6, C22⋊(C19⋊C6), (C22×D19)⋊2C3, SmallGroup(456,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C38 — D19⋊A4 |
Generators and relations for D19⋊A4
G = < a,b,c,d,e | a19=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, eae-1=a11, bc=cb, bd=db, ebe-1=a10b, ece-1=cd=dc, ede-1=c >
Character table of D19⋊A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 19A | 19B | 19C | 38A | 38B | 38C | 38D | 38E | 38F | 38G | 38H | 38I | |
| size | 1 | 3 | 19 | 57 | 76 | 76 | 76 | 76 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 3 | -1 | 3 | -1 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ8 | 3 | -1 | -3 | 1 | 0 | 0 | 0 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ9 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | orthogonal faithful |
| ρ10 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | orthogonal faithful |
| ρ11 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | orthogonal faithful |
| ρ12 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | orthogonal lifted from C19⋊C6 |
| ρ13 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | orthogonal faithful |
| ρ14 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | orthogonal faithful |
| ρ15 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | orthogonal faithful |
| ρ16 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | orthogonal faithful |
| ρ17 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | orthogonal lifted from C19⋊C6 |
| ρ18 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | orthogonal faithful |
| ρ19 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1917-ζ1916-ζ1914-ζ195-ζ193+ζ192 | ζ1915-ζ1913-ζ1910-ζ199-ζ196+ζ194 | -ζ1917-ζ1916+ζ1914+ζ195-ζ193-ζ192 | -ζ1918+ζ1912-ζ1911-ζ198+ζ197-ζ19 | -ζ1917+ζ1916-ζ1914-ζ195+ζ193-ζ192 | -ζ1915+ζ1913-ζ1910-ζ199+ζ196-ζ194 | -ζ1915-ζ1913+ζ1910+ζ199-ζ196-ζ194 | ζ1918-ζ1912-ζ1911-ζ198-ζ197+ζ19 | -ζ1918-ζ1912+ζ1911+ζ198-ζ197-ζ19 | orthogonal faithful |
| ρ20 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | orthogonal lifted from C19⋊C6 |
►On 76 points
Generators in S76
(1 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 54 55 56 57)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(37 38)(39 41)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(58 66)(59 65)(60 64)(61 63)(67 76)(68 75)(69 74)(70 73)(71 72)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 73)(21 74)(22 75)(23 76)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)
(1 38)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 58)(56 59)(57 60)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(20 60 42)(21 67 53)(22 74 45)(23 62 56)(24 69 48)(25 76 40)(26 64 51)(27 71 43)(28 59 54)(29 66 46)(30 73 57)(31 61 49)(32 68 41)(33 75 52)(34 63 44)(35 70 55)(36 58 47)(37 65 39)(38 72 50)
G:=sub<Sym(76)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)(39,41)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(58,66)(59,65)(60,64)(61,63)(67,76)(68,75)(69,74)(70,73)(71,72), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,73)(21,74)(22,75)(23,76)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72), (1,38)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,58)(56,59)(57,60), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(20,60,42)(21,67,53)(22,74,45)(23,62,56)(24,69,48)(25,76,40)(26,64,51)(27,71,43)(28,59,54)(29,66,46)(30,73,57)(31,61,49)(32,68,41)(33,75,52)(34,63,44)(35,70,55)(36,58,47)(37,65,39)(38,72,50)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)(39,41)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(58,66)(59,65)(60,64)(61,63)(67,76)(68,75)(69,74)(70,73)(71,72), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,73)(21,74)(22,75)(23,76)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72), (1,38)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,58)(56,59)(57,60), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(20,60,42)(21,67,53)(22,74,45)(23,62,56)(24,69,48)(25,76,40)(26,64,51)(27,71,43)(28,59,54)(29,66,46)(30,73,57)(31,61,49)(32,68,41)(33,75,52)(34,63,44)(35,70,55)(36,58,47)(37,65,39)(38,72,50) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(37,38),(39,41),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(58,66),(59,65),(60,64),(61,63),(67,76),(68,75),(69,74),(70,73),(71,72)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,73),(21,74),(22,75),(23,76),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72)], [(1,38),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,58),(56,59),(57,60)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(20,60,42),(21,67,53),(22,74,45),(23,62,56),(24,69,48),(25,76,40),(26,64,51),(27,71,43),(28,59,54),(29,66,46),(30,73,57),(31,61,49),(32,68,41),(33,75,52),(34,63,44),(35,70,55),(36,58,47),(37,65,39),(38,72,50)]])
Matrix representation of D19⋊A4 ►in GL6(𝔽229)
| 101 | 1 | 0 | 0 | 0 | 0 |
| 106 | 0 | 1 | 0 | 0 | 0 |
| 83 | 0 | 0 | 1 | 0 | 0 |
| 106 | 0 | 0 | 0 | 1 | 0 |
| 101 | 0 | 0 | 0 | 0 | 1 |
| 228 | 0 | 0 | 0 | 0 | 0 |
| 19 | 210 | 122 | 102 | 1 | 227 |
| 123 | 209 | 146 | 4 | 126 | 101 |
| 27 | 99 | 125 | 77 | 224 | 107 |
| 125 | 99 | 27 | 2 | 24 | 210 |
| 146 | 209 | 123 | 204 | 104 | 228 |
| 122 | 210 | 19 | 107 | 127 | 228 |
| 112 | 143 | 18 | 0 | 211 | 86 |
| 92 | 96 | 157 | 18 | 215 | 227 |
| 55 | 48 | 20 | 157 | 94 | 30 |
| 30 | 94 | 157 | 20 | 48 | 55 |
| 227 | 215 | 18 | 157 | 96 | 92 |
| 86 | 211 | 0 | 18 | 143 | 112 |
| 52 | 128 | 6 | 0 | 223 | 101 |
| 77 | 177 | 209 | 6 | 148 | 98 |
| 45 | 20 | 228 | 209 | 184 | 205 |
| 205 | 184 | 209 | 228 | 20 | 45 |
| 98 | 148 | 6 | 209 | 177 | 77 |
| 101 | 223 | 0 | 6 | 128 | 52 |
| 0 | 1 | 228 | 0 | 122 | 101 |
| 0 | 126 | 228 | 0 | 146 | 106 |
| 1 | 224 | 210 | 0 | 125 | 83 |
| 0 | 24 | 107 | 0 | 27 | 106 |
| 0 | 104 | 101 | 1 | 123 | 101 |
| 0 | 127 | 227 | 0 | 19 | 228 |
G:=sub<GL(6,GF(229))| [101,106,83,106,101,228,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[19,123,27,125,146,122,210,209,99,99,209,210,122,146,125,27,123,19,102,4,77,2,204,107,1,126,224,24,104,127,227,101,107,210,228,228],[112,92,55,30,227,86,143,96,48,94,215,211,18,157,20,157,18,0,0,18,157,20,157,18,211,215,94,48,96,143,86,227,30,55,92,112],[52,77,45,205,98,101,128,177,20,184,148,223,6,209,228,209,6,0,0,6,209,228,209,6,223,148,184,20,177,128,101,98,205,45,77,52],[0,0,1,0,0,0,1,126,224,24,104,127,228,228,210,107,101,227,0,0,0,0,1,0,122,146,125,27,123,19,101,106,83,106,101,228] >;
D19⋊A4 in GAP, Magma, Sage, TeX
D_{19}\rtimes A_4 % in TeX
G:=Group("D19:A4"); // GroupNames label
G:=SmallGroup(456,46);
// by ID
G=gap.SmallGroup(456,46);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-19,97,188,10804,2109]);
// Polycyclic
G:=Group<a,b,c,d,e|a^19=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^11,b*c=c*b,b*d=d*b,e*b*e^-1=a^10*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of D19⋊A4 in TeX
Character table of D19⋊A4 in TeX