metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C91⋊4C3, C7⋊2(C13⋊C3), C13⋊2(C7⋊C3), SmallGroup(273,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C91 — C91⋊4C3 |
Generators and relations for C91⋊4C3
G = < a,b | a91=b3=1, bab-1=a9 >
Smallest permutation representation of C91⋊4C3
►On 91 points
Generators in S91
(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)
(2 82 10)(3 72 19)(4 62 28)(5 52 37)(6 42 46)(7 32 55)(8 22 64)(9 12 73)(11 83 91)(13 63 18)(14 53 27)(15 43 36)(16 33 45)(17 23 54)(20 84 81)(21 74 90)(24 44 26)(25 34 35)(29 85 71)(30 75 80)(31 65 89)(38 86 61)(39 76 70)(40 66 79)(41 56 88)(47 87 51)(48 77 60)(49 67 69)(50 57 78)(58 68 59)
G:=sub<Sym(91)| (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), (2,82,10)(3,72,19)(4,62,28)(5,52,37)(6,42,46)(7,32,55)(8,22,64)(9,12,73)(11,83,91)(13,63,18)(14,53,27)(15,43,36)(16,33,45)(17,23,54)(20,84,81)(21,74,90)(24,44,26)(25,34,35)(29,85,71)(30,75,80)(31,65,89)(38,86,61)(39,76,70)(40,66,79)(41,56,88)(47,87,51)(48,77,60)(49,67,69)(50,57,78)(58,68,59)>;
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), (2,82,10)(3,72,19)(4,62,28)(5,52,37)(6,42,46)(7,32,55)(8,22,64)(9,12,73)(11,83,91)(13,63,18)(14,53,27)(15,43,36)(16,33,45)(17,23,54)(20,84,81)(21,74,90)(24,44,26)(25,34,35)(29,85,71)(30,75,80)(31,65,89)(38,86,61)(39,76,70)(40,66,79)(41,56,88)(47,87,51)(48,77,60)(49,67,69)(50,57,78)(58,68,59) );
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)], [(2,82,10),(3,72,19),(4,62,28),(5,52,37),(6,42,46),(7,32,55),(8,22,64),(9,12,73),(11,83,91),(13,63,18),(14,53,27),(15,43,36),(16,33,45),(17,23,54),(20,84,81),(21,74,90),(24,44,26),(25,34,35),(29,85,71),(30,75,80),(31,65,89),(38,86,61),(39,76,70),(40,66,79),(41,56,88),(47,87,51),(48,77,60),(49,67,69),(50,57,78),(58,68,59)]])
33 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 13A | 13B | 13C | 13D | 91A | ··· | 91X |
| order | 1 | 3 | 3 | 7 | 7 | 13 | 13 | 13 | 13 | 91 | ··· | 91 |
| size | 1 | 91 | 91 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
33 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C7⋊C3 | C13⋊C3 | C91⋊4C3 |
| kernel | C91⋊4C3 | C91 | C13 | C7 | C1 |
| # reps | 1 | 2 | 2 | 4 | 24 |
Matrix representation of C91⋊4C3 ►in GL3(𝔽547) generated by
| 372 | 450 | 9 |
| 9 | 135 | 462 |
| 462 | 424 | 204 |
| 1 | 0 | 0 |
| 365 | 390 | 181 |
| 97 | 182 | 156 |
G:=sub<GL(3,GF(547))| [372,9,462,450,135,424,9,462,204],[1,365,97,0,390,182,0,181,156] >;
C91⋊4C3 in GAP, Magma, Sage, TeX
C_{91}\rtimes_4C_3 % in TeX
G:=Group("C91:4C3"); // GroupNames label
G:=SmallGroup(273,4);
// by ID
G=gap.SmallGroup(273,4);
# by ID
G:=PCGroup([3,-3,-7,-13,73,569]);
// Polycyclic
G:=Group<a,b|a^91=b^3=1,b*a*b^-1=a^9>;
// generators/relations
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