metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C91⋊1C4, C13⋊Dic7, D13.D7, C7⋊(C13⋊C4), (C7×D13).1C2, SmallGroup(364,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C91 — C91⋊C4 |
Generators and relations for C91⋊C4
G = < a,b | a91=b4=1, bab-1=a34 >
Smallest permutation representation of C91⋊C4
►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 84 65 35)(3 76 38 69)(4 68 11 12)(5 60 75 46)(6 52 48 80)(7 44 21 23)(8 36 85 57)(9 28 58 91)(10 20 31 34)(13 87 41 45)(14 79)(15 71 78 22)(16 63 51 56)(17 55 24 90)(18 47 88 33)(19 39 61 67)(25 82 81 89)(26 74 54 32)(27 66)(29 50 64 43)(30 42 37 77)(40 53)(49 72 70 86)(59 83 73 62)
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), (2,84,65,35)(3,76,38,69)(4,68,11,12)(5,60,75,46)(6,52,48,80)(7,44,21,23)(8,36,85,57)(9,28,58,91)(10,20,31,34)(13,87,41,45)(14,79)(15,71,78,22)(16,63,51,56)(17,55,24,90)(18,47,88,33)(19,39,61,67)(25,82,81,89)(26,74,54,32)(27,66)(29,50,64,43)(30,42,37,77)(40,53)(49,72,70,86)(59,83,73,62)>;
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), (2,84,65,35)(3,76,38,69)(4,68,11,12)(5,60,75,46)(6,52,48,80)(7,44,21,23)(8,36,85,57)(9,28,58,91)(10,20,31,34)(13,87,41,45)(14,79)(15,71,78,22)(16,63,51,56)(17,55,24,90)(18,47,88,33)(19,39,61,67)(25,82,81,89)(26,74,54,32)(27,66)(29,50,64,43)(30,42,37,77)(40,53)(49,72,70,86)(59,83,73,62) );
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)], [(2,84,65,35),(3,76,38,69),(4,68,11,12),(5,60,75,46),(6,52,48,80),(7,44,21,23),(8,36,85,57),(9,28,58,91),(10,20,31,34),(13,87,41,45),(14,79),(15,71,78,22),(16,63,51,56),(17,55,24,90),(18,47,88,33),(19,39,61,67),(25,82,81,89),(26,74,54,32),(27,66),(29,50,64,43),(30,42,37,77),(40,53),(49,72,70,86),(59,83,73,62)]])
31 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 13A | 13B | 13C | 14A | 14B | 14C | 91A | ··· | 91R |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 13 | 13 | 13 | 14 | 14 | 14 | 91 | ··· | 91 |
| size | 1 | 13 | 91 | 91 | 2 | 2 | 2 | 4 | 4 | 4 | 26 | 26 | 26 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D7 | Dic7 | C13⋊C4 | C91⋊C4 |
| kernel | C91⋊C4 | C7×D13 | C91 | D13 | C13 | C7 | C1 |
| # reps | 1 | 1 | 2 | 3 | 3 | 3 | 18 |
Matrix representation of C91⋊C4 ►in GL4(𝔽1093) generated by
| 795 | 876 | 217 | 298 |
| 182 | 249 | 329 | 35 |
| 1010 | 935 | 589 | 710 |
| 1020 | 640 | 212 | 697 |
| 339 | 184 | 340 | 755 |
| 0 | 0 | 1 | 0 |
| 939 | 679 | 752 | 908 |
| 752 | 679 | 939 | 2 |
G:=sub<GL(4,GF(1093))| [795,182,1010,1020,876,249,935,640,217,329,589,212,298,35,710,697],[339,0,939,752,184,0,679,679,340,1,752,939,755,0,908,2] >;
C91⋊C4 in GAP, Magma, Sage, TeX
C_{91}\rtimes C_4 % in TeX
G:=Group("C91:C4"); // GroupNames label
G:=SmallGroup(364,6);
// by ID
G=gap.SmallGroup(364,6);
# by ID
G:=PCGroup([4,-2,-2,-7,-13,8,290,2243,2695]);
// Polycyclic
G:=Group<a,b|a^91=b^4=1,b*a*b^-1=a^34>;
// generators/relations
Export