metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C97⋊C3, SmallGroup(291,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C97 — C97⋊C3 |
Generators and relations for C97⋊C3
G = < a,b | a97=b3=1, bab-1=a61 >
Smallest permutation representation of C97⋊C3
►On 97 points: primitive
Generators in S97
(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)
(2 36 62)(3 71 26)(4 9 87)(5 44 51)(6 79 15)(7 17 76)(8 52 40)(10 25 65)(11 60 29)(12 95 90)(13 33 54)(14 68 18)(16 41 43)(19 49 32)(20 84 93)(21 22 57)(23 92 82)(24 30 46)(27 38 35)(28 73 96)(31 81 85)(34 89 74)(37 97 63)(39 70 88)(42 78 77)(45 86 66)(47 59 91)(48 94 55)(50 67 80)(53 75 69)(56 83 58)(61 64 72)
G:=sub<Sym(97)| (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), (2,36,62)(3,71,26)(4,9,87)(5,44,51)(6,79,15)(7,17,76)(8,52,40)(10,25,65)(11,60,29)(12,95,90)(13,33,54)(14,68,18)(16,41,43)(19,49,32)(20,84,93)(21,22,57)(23,92,82)(24,30,46)(27,38,35)(28,73,96)(31,81,85)(34,89,74)(37,97,63)(39,70,88)(42,78,77)(45,86,66)(47,59,91)(48,94,55)(50,67,80)(53,75,69)(56,83,58)(61,64,72)>;
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), (2,36,62)(3,71,26)(4,9,87)(5,44,51)(6,79,15)(7,17,76)(8,52,40)(10,25,65)(11,60,29)(12,95,90)(13,33,54)(14,68,18)(16,41,43)(19,49,32)(20,84,93)(21,22,57)(23,92,82)(24,30,46)(27,38,35)(28,73,96)(31,81,85)(34,89,74)(37,97,63)(39,70,88)(42,78,77)(45,86,66)(47,59,91)(48,94,55)(50,67,80)(53,75,69)(56,83,58)(61,64,72) );
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)], [(2,36,62),(3,71,26),(4,9,87),(5,44,51),(6,79,15),(7,17,76),(8,52,40),(10,25,65),(11,60,29),(12,95,90),(13,33,54),(14,68,18),(16,41,43),(19,49,32),(20,84,93),(21,22,57),(23,92,82),(24,30,46),(27,38,35),(28,73,96),(31,81,85),(34,89,74),(37,97,63),(39,70,88),(42,78,77),(45,86,66),(47,59,91),(48,94,55),(50,67,80),(53,75,69),(56,83,58),(61,64,72)]])
35 conjugacy classes
| class | 1 | 3A | 3B | 97A | ··· | 97AF |
| order | 1 | 3 | 3 | 97 | ··· | 97 |
| size | 1 | 97 | 97 | 3 | ··· | 3 |
35 irreducible representations
| dim | 1 | 1 | 3 |
| type | + | ||
| image | C1 | C3 | C97⋊C3 |
| kernel | C97⋊C3 | C97 | C1 |
| # reps | 1 | 2 | 32 |
Matrix representation of C97⋊C3 ►in GL3(𝔽1747) generated by
| 894 | 1 | 0 |
| 1549 | 0 | 1 |
| 1196 | 180 | 392 |
| 46 | 515 | 945 |
| 7 | 163 | 41 |
| 92 | 957 | 1538 |
G:=sub<GL(3,GF(1747))| [894,1549,1196,1,0,180,0,1,392],[46,7,92,515,163,957,945,41,1538] >;
C97⋊C3 in GAP, Magma, Sage, TeX
C_{97}\rtimes C_3 % in TeX
G:=Group("C97:C3"); // GroupNames label
G:=SmallGroup(291,1);
// by ID
G=gap.SmallGroup(291,1);
# by ID
G:=PCGroup([2,-3,-97,421]);
// Polycyclic
G:=Group<a,b|a^97=b^3=1,b*a*b^-1=a^61>;
// generators/relations
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