direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C38, C6⋊C38, C114⋊3C2, C57⋊4C22, C3⋊(C2×C38), SmallGroup(228,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C38 |
Generators and relations for S3×C38
G = < a,b,c | a38=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C38
►On 114 points
Generators in S114
(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 99 50)(2 100 51)(3 101 52)(4 102 53)(5 103 54)(6 104 55)(7 105 56)(8 106 57)(9 107 58)(10 108 59)(11 109 60)(12 110 61)(13 111 62)(14 112 63)(15 113 64)(16 114 65)(17 77 66)(18 78 67)(19 79 68)(20 80 69)(21 81 70)(22 82 71)(23 83 72)(24 84 73)(25 85 74)(26 86 75)(27 87 76)(28 88 39)(29 89 40)(30 90 41)(31 91 42)(32 92 43)(33 93 44)(34 94 45)(35 95 46)(36 96 47)(37 97 48)(38 98 49)
(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)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)
G:=sub<Sym(114)| (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), (1,99,50)(2,100,51)(3,101,52)(4,102,53)(5,103,54)(6,104,55)(7,105,56)(8,106,57)(9,107,58)(10,108,59)(11,109,60)(12,110,61)(13,111,62)(14,112,63)(15,113,64)(16,114,65)(17,77,66)(18,78,67)(19,79,68)(20,80,69)(21,81,70)(22,82,71)(23,83,72)(24,84,73)(25,85,74)(26,86,75)(27,87,76)(28,88,39)(29,89,40)(30,90,41)(31,91,42)(32,92,43)(33,93,44)(34,94,45)(35,95,46)(36,96,47)(37,97,48)(38,98,49), (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)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)>;
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)(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), (1,99,50)(2,100,51)(3,101,52)(4,102,53)(5,103,54)(6,104,55)(7,105,56)(8,106,57)(9,107,58)(10,108,59)(11,109,60)(12,110,61)(13,111,62)(14,112,63)(15,113,64)(16,114,65)(17,77,66)(18,78,67)(19,79,68)(20,80,69)(21,81,70)(22,82,71)(23,83,72)(24,84,73)(25,85,74)(26,86,75)(27,87,76)(28,88,39)(29,89,40)(30,90,41)(31,91,42)(32,92,43)(33,93,44)(34,94,45)(35,95,46)(36,96,47)(37,97,48)(38,98,49), (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)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106) );
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),(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)], [(1,99,50),(2,100,51),(3,101,52),(4,102,53),(5,103,54),(6,104,55),(7,105,56),(8,106,57),(9,107,58),(10,108,59),(11,109,60),(12,110,61),(13,111,62),(14,112,63),(15,113,64),(16,114,65),(17,77,66),(18,78,67),(19,79,68),(20,80,69),(21,81,70),(22,82,71),(23,83,72),(24,84,73),(25,85,74),(26,86,75),(27,87,76),(28,88,39),(29,89,40),(30,90,41),(31,91,42),(32,92,43),(33,93,44),(34,94,45),(35,95,46),(36,96,47),(37,97,48),(38,98,49)], [(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),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106)]])
S3×C38 is a maximal subgroup of
C57⋊D4 C19⋊D12
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38BB | 57A | ··· | 57R | 114A | ··· | 114R |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | S3 | D6 | S3×C19 | S3×C38 |
| kernel | S3×C38 | S3×C19 | C114 | D6 | S3 | C6 | C38 | C19 | C2 | C1 |
| # reps | 1 | 2 | 1 | 18 | 36 | 18 | 1 | 1 | 18 | 18 |
Matrix representation of S3×C38 ►in GL2(𝔽229) generated by
| 4 | 0 |
| 0 | 4 |
| 0 | 228 |
| 1 | 228 |
| 228 | 1 |
| 0 | 1 |
G:=sub<GL(2,GF(229))| [4,0,0,4],[0,1,228,228],[228,0,1,1] >;
S3×C38 in GAP, Magma, Sage, TeX
S_3\times C_{38} % in TeX
G:=Group("S3xC38"); // GroupNames label
G:=SmallGroup(228,13);
// by ID
G=gap.SmallGroup(228,13);
# by ID
G:=PCGroup([4,-2,-2,-19,-3,2435]);
// Polycyclic
G:=Group<a,b,c|a^38=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export