direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×S3×D19, C57⋊C23, C6⋊1D38, C38⋊1D6, C114⋊C22, D57⋊C22, D114⋊5C2, (S3×C38)⋊3C2, (C6×D19)⋊3C2, (S3×C19)⋊C22, C19⋊1(C22×S3), (C3×D19)⋊C22, C3⋊1(C22×D19), SmallGroup(456,47)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — C2×S3×D19 |
Generators and relations for C2×S3×D19
G = < a,b,c,d,e | a2=b3=c2=d19=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 792 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C23, D6, D6, C2×C6, C19, C22×S3, D19, D19, C38, C38, C57, D38, D38, C2×C38, S3×C19, C3×D19, D57, C114, C22×D19, S3×D19, C6×D19, S3×C38, D114, C2×S3×D19
Quotients: C1, C2, C22, S3, C23, D6, C22×S3, D19, D38, C22×D19, S3×D19, C2×S3×D19
►On 114 points
Generators in S114
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 58)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 77)(38 78)(39 113)(40 114)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)
(1 34 53)(2 35 54)(3 36 55)(4 37 56)(5 38 57)(6 20 39)(7 21 40)(8 22 41)(9 23 42)(10 24 43)(11 25 44)(12 26 45)(13 27 46)(14 28 47)(15 29 48)(16 30 49)(17 31 50)(18 32 51)(19 33 52)(58 92 107)(59 93 108)(60 94 109)(61 95 110)(62 77 111)(63 78 112)(64 79 113)(65 80 114)(66 81 96)(67 82 97)(68 83 98)(69 84 99)(70 85 100)(71 86 101)(72 87 102)(73 88 103)(74 89 104)(75 90 105)(76 91 106)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 58)(20 113)(21 114)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 77)(57 78)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 28)(21 27)(22 26)(23 25)(29 38)(30 37)(31 36)(32 35)(33 34)(39 47)(40 46)(41 45)(42 44)(48 57)(49 56)(50 55)(51 54)(52 53)(58 59)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(90 95)(91 94)(92 93)(96 100)(97 99)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(114)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,58)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,77)(38,78)(39,113)(40,114)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112), (1,34,53)(2,35,54)(3,36,55)(4,37,56)(5,38,57)(6,20,39)(7,21,40)(8,22,41)(9,23,42)(10,24,43)(11,25,44)(12,26,45)(13,27,46)(14,28,47)(15,29,48)(16,30,49)(17,31,50)(18,32,51)(19,33,52)(58,92,107)(59,93,108)(60,94,109)(61,95,110)(62,77,111)(63,78,112)(64,79,113)(65,80,114)(66,81,96)(67,82,97)(68,83,98)(69,84,99)(70,85,100)(71,86,101)(72,87,102)(73,88,103)(74,89,104)(75,90,105)(76,91,106), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,58)(20,113)(21,114)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,77)(57,78), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,28)(21,27)(22,26)(23,25)(29,38)(30,37)(31,36)(32,35)(33,34)(39,47)(40,46)(41,45)(42,44)(48,57)(49,56)(50,55)(51,54)(52,53)(58,59)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(90,95)(91,94)(92,93)(96,100)(97,99)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,58)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,77)(38,78)(39,113)(40,114)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112), (1,34,53)(2,35,54)(3,36,55)(4,37,56)(5,38,57)(6,20,39)(7,21,40)(8,22,41)(9,23,42)(10,24,43)(11,25,44)(12,26,45)(13,27,46)(14,28,47)(15,29,48)(16,30,49)(17,31,50)(18,32,51)(19,33,52)(58,92,107)(59,93,108)(60,94,109)(61,95,110)(62,77,111)(63,78,112)(64,79,113)(65,80,114)(66,81,96)(67,82,97)(68,83,98)(69,84,99)(70,85,100)(71,86,101)(72,87,102)(73,88,103)(74,89,104)(75,90,105)(76,91,106), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,58)(20,113)(21,114)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,77)(57,78), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,28)(21,27)(22,26)(23,25)(29,38)(30,37)(31,36)(32,35)(33,34)(39,47)(40,46)(41,45)(42,44)(48,57)(49,56)(50,55)(51,54)(52,53)(58,59)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(90,95)(91,94)(92,93)(96,100)(97,99)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,58),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,77),(38,78),(39,113),(40,114),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112)], [(1,34,53),(2,35,54),(3,36,55),(4,37,56),(5,38,57),(6,20,39),(7,21,40),(8,22,41),(9,23,42),(10,24,43),(11,25,44),(12,26,45),(13,27,46),(14,28,47),(15,29,48),(16,30,49),(17,31,50),(18,32,51),(19,33,52),(58,92,107),(59,93,108),(60,94,109),(61,95,110),(62,77,111),(63,78,112),(64,79,113),(65,80,114),(66,81,96),(67,82,97),(68,83,98),(69,84,99),(70,85,100),(71,86,101),(72,87,102),(73,88,103),(74,89,104),(75,90,105),(76,91,106)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,58),(20,113),(21,114),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,77),(57,78)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,28),(21,27),(22,26),(23,25),(29,38),(30,37),(31,36),(32,35),(33,34),(39,47),(40,46),(41,45),(42,44),(48,57),(49,56),(50,55),(51,54),(52,53),(58,59),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(90,95),(91,94),(92,93),(96,100),(97,99),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 57A | ··· | 57I | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 3 | 3 | 19 | 19 | 57 | 57 | 2 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D19 | D38 | D38 | S3×D19 | C2×S3×D19 |
| kernel | C2×S3×D19 | S3×D19 | C6×D19 | S3×C38 | D114 | D38 | D19 | C38 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 9 | 18 | 9 | 9 | 9 |
Matrix representation of C2×S3×D19 ►in GL4(𝔽229) generated by
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 228 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 227 | 91 |
| 0 | 0 | 5 | 1 |
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 5 | 1 |
| 126 | 1 | 0 | 0 |
| 192 | 196 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 86 | 54 | 0 | 0 |
| 3 | 143 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(229))| [228,0,0,0,0,228,0,0,0,0,228,0,0,0,0,228],[1,0,0,0,0,1,0,0,0,0,227,5,0,0,91,1],[228,0,0,0,0,228,0,0,0,0,228,5,0,0,0,1],[126,192,0,0,1,196,0,0,0,0,1,0,0,0,0,1],[86,3,0,0,54,143,0,0,0,0,1,0,0,0,0,1] >;
C2×S3×D19 in GAP, Magma, Sage, TeX
C_2\times S_3\times D_{19} % in TeX
G:=Group("C2xS3xD19"); // GroupNames label
G:=SmallGroup(456,47);
// by ID
G=gap.SmallGroup(456,47);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,168,10804]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^19=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations