metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C13⋊2M4(2), Dic13.3C4, Dic13.7C22, C13⋊C8⋊2C2, C26.6(C2×C4), (C2×C26).2C4, C22.(C13⋊C4), (C2×Dic13).5C2, C2.6(C2×C13⋊C4), SmallGroup(208,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13⋊M4(2)
G = < a,b,c | a13=b8=c2=1, bab-1=a5, ac=ca, cbc=b5 >
Character table of C13⋊M4(2)
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | 26B | 26C | 26D | 26E | 26F | 26G | 26H | 26I | |
| size | 1 | 1 | 2 | 13 | 13 | 26 | 26 | 26 | 26 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ139-ζ137-ζ136-ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | orthogonal lifted from C2×C13⋊C4 |
| ρ12 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ13 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ1312-ζ138-ζ135-ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | orthogonal lifted from C2×C13⋊C4 |
| ρ14 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ15 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1311-ζ1310-ζ133-ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | orthogonal lifted from C2×C13⋊C4 |
| ρ16 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | -ζ1311+ζ1310+ζ133-ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ139+ζ137+ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | ζ1312-ζ138-ζ135+ζ13 | ζ1311-ζ1310-ζ133+ζ132 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1311-ζ1310-ζ133+ζ132 | -ζ139+ζ137+ζ136-ζ134 | ζ1312-ζ138-ζ135+ζ13 | ζ139-ζ137-ζ136+ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ139-ζ137-ζ136+ζ134 | ζ1312-ζ138-ζ135+ζ13 | ζ1311-ζ1310-ζ133+ζ132 | -ζ1312+ζ138+ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ139+ζ137+ζ136-ζ134 | symplectic faithful, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312-ζ138-ζ135+ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ139+ζ137+ζ136-ζ134 | ζ1311-ζ1310-ζ133+ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ1312+ζ138+ζ135-ζ13 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | -ζ1312+ζ138+ζ135-ζ13 | ζ1311-ζ1310-ζ133+ζ132 | ζ139-ζ137-ζ136+ζ134 | -ζ1311+ζ1310+ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139+ζ137+ζ136-ζ134 | ζ1312-ζ138-ζ135+ζ13 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | -ζ139+ζ137+ζ136-ζ134 | -ζ1312+ζ138+ζ135-ζ13 | -ζ1311+ζ1310+ζ133-ζ132 | ζ1312-ζ138-ζ135+ζ13 | -ζ1311-ζ1310-ζ133-ζ132 | -ζ139-ζ137-ζ136-ζ134 | -ζ1312-ζ138-ζ135-ζ13 | ζ1311-ζ1310-ζ133+ζ132 | ζ139-ζ137-ζ136+ζ134 | symplectic faithful, Schur index 2 |
►On 104 points
Generators in S104
(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 99 100 101 102 103 104)
(1 84 27 74 14 103 40 56)(2 79 39 66 15 98 52 61)(3 87 38 71 16 93 51 53)(4 82 37 76 17 101 50 58)(5 90 36 68 18 96 49 63)(6 85 35 73 19 104 48 55)(7 80 34 78 20 99 47 60)(8 88 33 70 21 94 46 65)(9 83 32 75 22 102 45 57)(10 91 31 67 23 97 44 62)(11 86 30 72 24 92 43 54)(12 81 29 77 25 100 42 59)(13 89 28 69 26 95 41 64)
(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 66)(62 67)(63 68)(64 69)(65 70)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 92)(87 93)(88 94)(89 95)(90 96)(91 97)
G:=sub<Sym(104)| (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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,84,27,74,14,103,40,56)(2,79,39,66,15,98,52,61)(3,87,38,71,16,93,51,53)(4,82,37,76,17,101,50,58)(5,90,36,68,18,96,49,63)(6,85,35,73,19,104,48,55)(7,80,34,78,20,99,47,60)(8,88,33,70,21,94,46,65)(9,83,32,75,22,102,45,57)(10,91,31,67,23,97,44,62)(11,86,30,72,24,92,43,54)(12,81,29,77,25,100,42,59)(13,89,28,69,26,95,41,64), (53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,66)(62,67)(63,68)(64,69)(65,70)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,92)(87,93)(88,94)(89,95)(90,96)(91,97)>;
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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,84,27,74,14,103,40,56)(2,79,39,66,15,98,52,61)(3,87,38,71,16,93,51,53)(4,82,37,76,17,101,50,58)(5,90,36,68,18,96,49,63)(6,85,35,73,19,104,48,55)(7,80,34,78,20,99,47,60)(8,88,33,70,21,94,46,65)(9,83,32,75,22,102,45,57)(10,91,31,67,23,97,44,62)(11,86,30,72,24,92,43,54)(12,81,29,77,25,100,42,59)(13,89,28,69,26,95,41,64), (53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,66)(62,67)(63,68)(64,69)(65,70)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,92)(87,93)(88,94)(89,95)(90,96)(91,97) );
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),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,84,27,74,14,103,40,56),(2,79,39,66,15,98,52,61),(3,87,38,71,16,93,51,53),(4,82,37,76,17,101,50,58),(5,90,36,68,18,96,49,63),(6,85,35,73,19,104,48,55),(7,80,34,78,20,99,47,60),(8,88,33,70,21,94,46,65),(9,83,32,75,22,102,45,57),(10,91,31,67,23,97,44,62),(11,86,30,72,24,92,43,54),(12,81,29,77,25,100,42,59),(13,89,28,69,26,95,41,64)], [(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,66),(62,67),(63,68),(64,69),(65,70),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,104),(86,92),(87,93),(88,94),(89,95),(90,96),(91,97)]])
C13⋊M4(2) is a maximal subgroup of
Dic13.D4 Dic13.4D4 D13⋊M4(2) Dic26.C4
C13⋊M4(2) is a maximal quotient of C26.C42 Dic13⋊C8 C26.M4(2)
Matrix representation of C13⋊M4(2) ►in GL6(𝔽313)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 312 | 101 | 28 | 101 |
| 25 | 311 | 0 | 0 | 0 | 0 |
| 12 | 288 | 0 | 0 | 0 | 0 |
| 0 | 0 | 305 | 251 | 164 | 74 |
| 0 | 0 | 307 | 233 | 256 | 303 |
| 0 | 0 | 78 | 109 | 63 | 290 |
| 0 | 0 | 187 | 132 | 39 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 25 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,312,0,0,1,0,0,101,0,0,0,1,0,28,0,0,0,0,1,101],[25,12,0,0,0,0,311,288,0,0,0,0,0,0,305,307,78,187,0,0,251,233,109,132,0,0,164,256,63,39,0,0,74,303,290,25],[1,25,0,0,0,0,0,312,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C13⋊M4(2) in GAP, Magma, Sage, TeX
C_{13}\rtimes M_4(2) % in TeX
G:=Group("C13:M4(2)"); // GroupNames label
G:=SmallGroup(208,33);
// by ID
G=gap.SmallGroup(208,33);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,42,3204,1214]);
// Polycyclic
G:=Group<a,b,c|a^13=b^8=c^2=1,b*a*b^-1=a^5,a*c=c*a,c*b*c=b^5>;
// generators/relations
Export
Subgroup lattice of C13⋊M4(2) in TeX
Character table of C13⋊M4(2) in TeX