direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×C13⋊C4, C13⋊C28, C91⋊2C4, D13.C14, (C7×D13).2C2, SmallGroup(364,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C7×C13⋊C4 |
Generators and relations for C7×C13⋊C4
G = < a,b,c | a7=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C7×C13⋊C4
►On 91 points
Generators in S91
(1 79 66 53 40 27 14)(2 80 67 54 41 28 15)(3 81 68 55 42 29 16)(4 82 69 56 43 30 17)(5 83 70 57 44 31 18)(6 84 71 58 45 32 19)(7 85 72 59 46 33 20)(8 86 73 60 47 34 21)(9 87 74 61 48 35 22)(10 88 75 62 49 36 23)(11 89 76 63 50 37 24)(12 90 77 64 51 38 25)(13 91 78 65 52 39 26)
(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 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)(67 74 78 71)(68 69 77 76)(70 72 75 73)(80 87 91 84)(81 82 90 89)(83 85 88 86)
G:=sub<Sym(91)| (1,79,66,53,40,27,14)(2,80,67,54,41,28,15)(3,81,68,55,42,29,16)(4,82,69,56,43,30,17)(5,83,70,57,44,31,18)(6,84,71,58,45,32,19)(7,85,72,59,46,33,20)(8,86,73,60,47,34,21)(9,87,74,61,48,35,22)(10,88,75,62,49,36,23)(11,89,76,63,50,37,24)(12,90,77,64,51,38,25)(13,91,78,65,52,39,26), (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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)(67,74,78,71)(68,69,77,76)(70,72,75,73)(80,87,91,84)(81,82,90,89)(83,85,88,86)>;
G:=Group( (1,79,66,53,40,27,14)(2,80,67,54,41,28,15)(3,81,68,55,42,29,16)(4,82,69,56,43,30,17)(5,83,70,57,44,31,18)(6,84,71,58,45,32,19)(7,85,72,59,46,33,20)(8,86,73,60,47,34,21)(9,87,74,61,48,35,22)(10,88,75,62,49,36,23)(11,89,76,63,50,37,24)(12,90,77,64,51,38,25)(13,91,78,65,52,39,26), (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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)(67,74,78,71)(68,69,77,76)(70,72,75,73)(80,87,91,84)(81,82,90,89)(83,85,88,86) );
G=PermutationGroup([[(1,79,66,53,40,27,14),(2,80,67,54,41,28,15),(3,81,68,55,42,29,16),(4,82,69,56,43,30,17),(5,83,70,57,44,31,18),(6,84,71,58,45,32,19),(7,85,72,59,46,33,20),(8,86,73,60,47,34,21),(9,87,74,61,48,35,22),(10,88,75,62,49,36,23),(11,89,76,63,50,37,24),(12,90,77,64,51,38,25),(13,91,78,65,52,39,26)], [(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)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(15,22,26,19),(16,17,25,24),(18,20,23,21),(28,35,39,32),(29,30,38,37),(31,33,36,34),(41,48,52,45),(42,43,51,50),(44,46,49,47),(54,61,65,58),(55,56,64,63),(57,59,62,60),(67,74,78,71),(68,69,77,76),(70,72,75,73),(80,87,91,84),(81,82,90,89),(83,85,88,86)]])
49 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7F | 13A | 13B | 13C | 14A | ··· | 14F | 28A | ··· | 28L | 91A | ··· | 91R |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 13 | 13 | 13 | 14 | ··· | 14 | 28 | ··· | 28 | 91 | ··· | 91 |
| size | 1 | 13 | 13 | 13 | 1 | ··· | 1 | 4 | 4 | 4 | 13 | ··· | 13 | 13 | ··· | 13 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | C13⋊C4 | C7×C13⋊C4 |
| kernel | C7×C13⋊C4 | C7×D13 | C91 | C13⋊C4 | D13 | C13 | C7 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 3 | 18 |
Matrix representation of C7×C13⋊C4 ►in GL4(𝔽1093) generated by
| 243 | 0 | 0 | 0 |
| 0 | 243 | 0 | 0 |
| 0 | 0 | 243 | 0 |
| 0 | 0 | 0 | 243 |
| 569 | 523 | 756 | 1092 |
| 1 | 0 | 0 | 907 |
| 0 | 1 | 0 | 1091 |
| 0 | 0 | 1 | 338 |
| 524 | 186 | 1092 | 906 |
| 233 | 817 | 617 | 673 |
| 234 | 708 | 799 | 905 |
| 0 | 572 | 862 | 46 |
G:=sub<GL(4,GF(1093))| [243,0,0,0,0,243,0,0,0,0,243,0,0,0,0,243],[569,1,0,0,523,0,1,0,756,0,0,1,1092,907,1091,338],[524,233,234,0,186,817,708,572,1092,617,799,862,906,673,905,46] >;
C7×C13⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_{13}\rtimes C_4 % in TeX
G:=Group("C7xC13:C4"); // GroupNames label
G:=SmallGroup(364,5);
// by ID
G=gap.SmallGroup(364,5);
# by ID
G:=PCGroup([4,-2,-7,-2,-13,56,3587,395]);
// Polycyclic
G:=Group<a,b,c|a^7=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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