non-abelian, soluble, monomial
Aliases: C19⋊S4, A4⋊D19, C22⋊D57, (C2×C38)⋊2S3, (A4×C19)⋊1C2, SmallGroup(456,43)
Series: Derived ►Chief ►Lower central ►Upper central
| A4×C19 — C19⋊S4 |
Generators and relations for C19⋊S4
G = < a,b,c,d,e | a19=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Smallest permutation representation of C19⋊S4
►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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)
(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 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)
(20 39 68)(21 40 69)(22 41 70)(23 42 71)(24 43 72)(25 44 73)(26 45 74)(27 46 75)(28 47 76)(29 48 58)(30 49 59)(31 50 60)(32 51 61)(33 52 62)(34 53 63)(35 54 64)(36 55 65)(37 56 66)(38 57 67)
(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 60)(32 59)(33 58)(34 76)(35 75)(36 74)(37 73)(38 72)(39 42)(40 41)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)
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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67), (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)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67), (20,39,68)(21,40,69)(22,41,70)(23,42,71)(24,43,72)(25,44,73)(26,45,74)(27,46,75)(28,47,76)(29,48,58)(30,49,59)(31,50,60)(32,51,61)(33,52,62)(34,53,63)(35,54,64)(36,55,65)(37,56,66)(38,57,67), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,76)(35,75)(36,74)(37,73)(38,72)(39,42)(40,41)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)>;
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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67), (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)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67), (20,39,68)(21,40,69)(22,41,70)(23,42,71)(24,43,72)(25,44,73)(26,45,74)(27,46,75)(28,47,76)(29,48,58)(30,49,59)(31,50,60)(32,51,61)(33,52,62)(34,53,63)(35,54,64)(36,55,65)(37,56,66)(38,57,67), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,76)(35,75)(36,74)(37,73)(38,72)(39,42)(40,41)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51) );
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)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67)], [(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),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67)], [(20,39,68),(21,40,69),(22,41,70),(23,42,71),(24,43,72),(25,44,73),(26,45,74),(27,46,75),(28,47,76),(29,48,58),(30,49,59),(31,50,60),(32,51,61),(33,52,62),(34,53,63),(35,54,64),(36,55,65),(37,56,66),(38,57,67)], [(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,60),(32,59),(33,58),(34,76),(35,75),(36,74),(37,73),(38,72),(39,42),(40,41),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4 | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R |
| order | 1 | 2 | 2 | 3 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 |
| size | 1 | 3 | 114 | 8 | 114 | 2 | ··· | 2 | 6 | ··· | 6 | 8 | ··· | 8 |
41 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 3 | 6 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D19 | D57 | S4 | C19⋊S4 |
| kernel | C19⋊S4 | A4×C19 | C2×C38 | A4 | C22 | C19 | C1 |
| # reps | 1 | 1 | 1 | 9 | 18 | 2 | 9 |
Matrix representation of C19⋊S4 ►in GL5(𝔽229)
| 108 | 57 | 0 | 0 | 0 |
| 172 | 165 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 228 | 1 |
| 0 | 0 | 0 | 228 | 0 |
| 0 | 0 | 1 | 228 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 228 |
| 0 | 0 | 1 | 0 | 228 |
| 0 | 0 | 0 | 0 | 228 |
| 0 | 228 | 0 | 0 | 0 |
| 1 | 228 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 228 | 0 | 0 | 0 | 0 |
| 228 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(229))| [108,172,0,0,0,57,165,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,228,228,228,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,228,228,228],[0,1,0,0,0,228,228,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[228,228,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
C19⋊S4 in GAP, Magma, Sage, TeX
C_{19}\rtimes S_4 % in TeX
G:=Group("C19:S4"); // GroupNames label
G:=SmallGroup(456,43);
// by ID
G=gap.SmallGroup(456,43);
# by ID
G:=PCGroup([5,-2,-3,-19,-2,2,41,1622,4563,2288,2854,4284]);
// Polycyclic
G:=Group<a,b,c,d,e|a^19=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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