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## G = C13⋊M4(2)  order 208 = 24·13

### The semidirect product of C13 and M4(2) acting via M4(2)/C22=C4

Aliases: C132M4(2), Dic13.3C4, Dic13.7C22, C13⋊C82C2, 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

 Derived series C1 — C26 — C13⋊M4(2)
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — C13⋊M4(2)
 Lower central C13 — C26 — C13⋊M4(2)
 Upper central C1 — C2 — C22

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

Smallest permutation representation of C13⋊M4(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`

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