direct product, metabelian, soluble, monomial, A-group
Aliases: C2xC19:A4, C38:A4, C19:2(C2xA4), C23:(C19:C3), (C2xC38):7C6, (C22xC38):3C3, C22:(C2xC19:C3), SmallGroup(456,50)
Series: Derived ►Chief ►Lower central ►Upper central
| C2xC38 — C2xC19:A4 |
Generators and relations for C2xC19:A4
G = < a,b,c,d,e | a2=b19=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b11, ece-1=cd=dc, ede-1=c >
Quotients: C1, C2, C3, C6, A4, C2xA4, C19:C3, C2xC19:C3, C19:A4, C2xC19:A4
Smallest permutation representation of C2xC19:A4
►On 114 points
Generators in S114
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 58)(53 59)(54 60)(55 61)(56 62)(57 63)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 96)(93 97)(94 98)(95 99)
(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)(77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95)(96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 58)(53 59)(54 60)(55 61)(56 62)(57 63)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 96)(93 97)(94 98)(95 99)
(1 92 43)(2 80 54)(3 87 46)(4 94 57)(5 82 49)(6 89 41)(7 77 52)(8 84 44)(9 91 55)(10 79 47)(11 86 39)(12 93 50)(13 81 42)(14 88 53)(15 95 45)(16 83 56)(17 90 48)(18 78 40)(19 85 51)(20 96 68)(21 103 60)(22 110 71)(23 98 63)(24 105 74)(25 112 66)(26 100 58)(27 107 69)(28 114 61)(29 102 72)(30 109 64)(31 97 75)(32 104 67)(33 111 59)(34 99 70)(35 106 62)(36 113 73)(37 101 65)(38 108 76)
G:=sub<Sym(114)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (1,92,43)(2,80,54)(3,87,46)(4,94,57)(5,82,49)(6,89,41)(7,77,52)(8,84,44)(9,91,55)(10,79,47)(11,86,39)(12,93,50)(13,81,42)(14,88,53)(15,95,45)(16,83,56)(17,90,48)(18,78,40)(19,85,51)(20,96,68)(21,103,60)(22,110,71)(23,98,63)(24,105,74)(25,112,66)(26,100,58)(27,107,69)(28,114,61)(29,102,72)(30,109,64)(31,97,75)(32,104,67)(33,111,59)(34,99,70)(35,106,62)(36,113,73)(37,101,65)(38,108,76)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (1,92,43)(2,80,54)(3,87,46)(4,94,57)(5,82,49)(6,89,41)(7,77,52)(8,84,44)(9,91,55)(10,79,47)(11,86,39)(12,93,50)(13,81,42)(14,88,53)(15,95,45)(16,83,56)(17,90,48)(18,78,40)(19,85,51)(20,96,68)(21,103,60)(22,110,71)(23,98,63)(24,105,74)(25,112,66)(26,100,58)(27,107,69)(28,114,61)(29,102,72)(30,109,64)(31,97,75)(32,104,67)(33,111,59)(34,99,70)(35,106,62)(36,113,73)(37,101,65)(38,108,76) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,58),(53,59),(54,60),(55,61),(56,62),(57,63),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,96),(93,97),(94,98),(95,99)], [(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),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,58),(53,59),(54,60),(55,61),(56,62),(57,63)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,96),(93,97),(94,98),(95,99)], [(1,92,43),(2,80,54),(3,87,46),(4,94,57),(5,82,49),(6,89,41),(7,77,52),(8,84,44),(9,91,55),(10,79,47),(11,86,39),(12,93,50),(13,81,42),(14,88,53),(15,95,45),(16,83,56),(17,90,48),(18,78,40),(19,85,51),(20,96,68),(21,103,60),(22,110,71),(23,98,63),(24,105,74),(25,112,66),(26,100,58),(27,107,69),(28,114,61),(29,102,72),(30,109,64),(31,97,75),(32,104,67),(33,111,59),(34,99,70),(35,106,62),(36,113,73),(37,101,65),(38,108,76)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 19A | ··· | 19F | 38A | ··· | 38AP |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 |
| size | 1 | 1 | 3 | 3 | 76 | 76 | 76 | 76 | 3 | ··· | 3 | 3 | ··· | 3 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | A4 | C2xA4 | C19:C3 | C2xC19:C3 | C19:A4 | C2xC19:A4 |
| kernel | C2xC19:A4 | C19:A4 | C22xC38 | C2xC38 | C38 | C19 | C23 | C22 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 18 | 18 |
Matrix representation of C2xC19:A4 ►in GL3(F229) generated by
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 44 | 115 | 6 |
| 0 | 17 | 0 |
| 0 | 0 | 214 |
| 228 | 0 | 124 |
| 0 | 228 | 0 |
| 0 | 0 | 1 |
| 228 | 212 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 228 |
| 107 | 115 | 113 |
| 0 | 0 | 1 |
| 131 | 83 | 122 |
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[44,0,0,115,17,0,6,0,214],[228,0,0,0,228,0,124,0,1],[228,0,0,212,1,0,0,0,228],[107,0,131,115,0,83,113,1,122] >;
C2xC19:A4 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes A_4 % in TeX
G:=Group("C2xC19:A4"); // GroupNames label
G:=SmallGroup(456,50);
// by ID
G=gap.SmallGroup(456,50);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-19,97,188,2109]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^19=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^11,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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