direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C49⋊C3, C98⋊C3, C49⋊2C6, C14.(C7⋊C3), C7.(C2×C7⋊C3), SmallGroup(294,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C2×C49⋊C3 |
Generators and relations for C2×C49⋊C3
G = < a,b,c | a2=b49=c3=1, ab=ba, ac=ca, cbc-1=b18 >
Smallest permutation representation of C2×C49⋊C3
►On 98 points
Generators in S98
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)
(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)
(2 31 19)(3 12 37)(4 42 6)(5 23 24)(7 34 11)(8 15 29)(9 45 47)(10 26 16)(13 18 21)(14 48 39)(17 40 44)(20 32 49)(22 43 36)(25 35 41)(27 46 28)(30 38 33)(50 69 51)(52 80 87)(53 61 56)(54 91 74)(55 72 92)(57 83 79)(58 64 97)(59 94 66)(60 75 84)(62 86 71)(63 67 89)(65 78 76)(68 70 81)(77 95 96)(82 98 88)(85 90 93)
G:=sub<Sym(98)| (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72), (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), (2,31,19)(3,12,37)(4,42,6)(5,23,24)(7,34,11)(8,15,29)(9,45,47)(10,26,16)(13,18,21)(14,48,39)(17,40,44)(20,32,49)(22,43,36)(25,35,41)(27,46,28)(30,38,33)(50,69,51)(52,80,87)(53,61,56)(54,91,74)(55,72,92)(57,83,79)(58,64,97)(59,94,66)(60,75,84)(62,86,71)(63,67,89)(65,78,76)(68,70,81)(77,95,96)(82,98,88)(85,90,93)>;
G:=Group( (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72), (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), (2,31,19)(3,12,37)(4,42,6)(5,23,24)(7,34,11)(8,15,29)(9,45,47)(10,26,16)(13,18,21)(14,48,39)(17,40,44)(20,32,49)(22,43,36)(25,35,41)(27,46,28)(30,38,33)(50,69,51)(52,80,87)(53,61,56)(54,91,74)(55,72,92)(57,83,79)(58,64,97)(59,94,66)(60,75,84)(62,86,71)(63,67,89)(65,78,76)(68,70,81)(77,95,96)(82,98,88)(85,90,93) );
G=PermutationGroup([[(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,72)], [(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)], [(2,31,19),(3,12,37),(4,42,6),(5,23,24),(7,34,11),(8,15,29),(9,45,47),(10,26,16),(13,18,21),(14,48,39),(17,40,44),(20,32,49),(22,43,36),(25,35,41),(27,46,28),(30,38,33),(50,69,51),(52,80,87),(53,61,56),(54,91,74),(55,72,92),(57,83,79),(58,64,97),(59,94,66),(60,75,84),(62,86,71),(63,67,89),(65,78,76),(68,70,81),(77,95,96),(82,98,88),(85,90,93)]])
38 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 14A | 14B | 49A | ··· | 49N | 98A | ··· | 98N |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 14 | 14 | 49 | ··· | 49 | 98 | ··· | 98 |
| size | 1 | 1 | 49 | 49 | 49 | 49 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7⋊C3 | C2×C7⋊C3 | C49⋊C3 | C2×C49⋊C3 |
| kernel | C2×C49⋊C3 | C49⋊C3 | C98 | C49 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 |
Matrix representation of C2×C49⋊C3 ►in GL4(𝔽883) generated by
| 882 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 600 | 644 | 561 |
| 0 | 561 | 286 | 652 |
| 0 | 652 | 67 | 23 |
| 337 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 521 | 882 | 882 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(883))| [882,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,600,561,652,0,644,286,67,0,561,652,23],[337,0,0,0,0,1,521,0,0,0,882,1,0,0,882,0] >;
C2×C49⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{49}\rtimes C_3 % in TeX
G:=Group("C2xC49:C3"); // GroupNames label
G:=SmallGroup(294,2);
// by ID
G=gap.SmallGroup(294,2);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,330,178,679]);
// Polycyclic
G:=Group<a,b,c|a^2=b^49=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
// generators/relations
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