Copied to
clipboard

## G = C26.A4order 312 = 23·3·13

### The non-split extension by C26 of A4 acting via A4/C22=C3

Aliases: C26.A4, C13⋊SL2(𝔽3), Q8⋊(C13⋊C3), C2.(C13⋊A4), (Q8×C13)⋊2C3, SmallGroup(312,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C13 — C26.A4
 Chief series C1 — C2 — C26 — Q8×C13 — C26.A4
 Lower central Q8×C13 — C26.A4
 Upper central C1 — C2

Generators and relations for C26.A4
G = < a,b,c,d | a26=d3=1, b2=c2=a13, ab=ba, ac=ca, dad-1=a9, cbc-1=a13b, dbd-1=a13bc, dcd-1=b >

Character table of C26.A4

 class 1 2 3A 3B 4 6A 6B 13A 13B 13C 13D 26A 26B 26C 26D 52A 52B 52C 52D 52E 52F 52G 52H 52I 52J 52K 52L size 1 1 52 52 6 52 52 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 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 ζ32 ζ3 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 1 ζ3 ζ32 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 2 -2 -1 -1 0 1 1 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ5 2 -2 ζ65 ζ6 0 ζ32 ζ3 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from SL2(𝔽3) ρ6 2 -2 ζ6 ζ65 0 ζ3 ζ32 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from SL2(𝔽3) ρ7 3 3 0 0 -1 0 0 3 3 3 3 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ8 3 3 0 0 -1 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ1311-ζ138-ζ137 -ζ1311+ζ138-ζ137 -ζ136+ζ135-ζ132 -ζ136-ζ135+ζ132 ζ1312-ζ1310-ζ134 ζ139-ζ133-ζ13 ζ136-ζ135-ζ132 -ζ139+ζ133-ζ13 -ζ1312+ζ1310-ζ134 -ζ1311-ζ138+ζ137 -ζ1312-ζ1310+ζ134 -ζ139-ζ133+ζ13 complex lifted from C13⋊A4 ρ9 3 3 0 0 3 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 complex lifted from C13⋊C3 ρ10 3 3 0 0 -1 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 -ζ139-ζ133+ζ13 ζ139-ζ133-ζ13 -ζ1312-ζ1310+ζ134 ζ1312-ζ1310-ζ134 -ζ1311-ζ138+ζ137 -ζ136-ζ135+ζ132 -ζ1312+ζ1310-ζ134 -ζ136+ζ135-ζ132 -ζ1311+ζ138-ζ137 -ζ139+ζ133-ζ13 ζ1311-ζ138-ζ137 ζ136-ζ135-ζ132 complex lifted from C13⋊A4 ρ11 3 3 0 0 3 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ12 3 3 0 0 -1 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1312-ζ1310-ζ134 -ζ1312-ζ1310+ζ134 ζ139-ζ133-ζ13 -ζ139-ζ133+ζ13 ζ136-ζ135-ζ132 ζ1311-ζ138-ζ137 -ζ139+ζ133-ζ13 -ζ1311+ζ138-ζ137 -ζ136+ζ135-ζ132 -ζ1312+ζ1310-ζ134 -ζ136-ζ135+ζ132 -ζ1311-ζ138+ζ137 complex lifted from C13⋊A4 ρ13 3 3 0 0 -1 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 -ζ136+ζ135-ζ132 ζ136-ζ135-ζ132 -ζ1311-ζ138+ζ137 -ζ1311+ζ138-ζ137 ζ139-ζ133-ζ13 -ζ1312+ζ1310-ζ134 ζ1311-ζ138-ζ137 ζ1312-ζ1310-ζ134 -ζ139-ζ133+ζ13 -ζ136-ζ135+ζ132 -ζ139+ζ133-ζ13 -ζ1312-ζ1310+ζ134 complex lifted from C13⋊A4 ρ14 3 3 0 0 3 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ15 3 3 0 0 -1 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 -ζ1311+ζ138-ζ137 -ζ1311-ζ138+ζ137 ζ136-ζ135-ζ132 -ζ136+ζ135-ζ132 -ζ1312-ζ1310+ζ134 -ζ139+ζ133-ζ13 -ζ136-ζ135+ζ132 -ζ139-ζ133+ζ13 ζ1312-ζ1310-ζ134 ζ1311-ζ138-ζ137 -ζ1312+ζ1310-ζ134 ζ139-ζ133-ζ13 complex lifted from C13⋊A4 ρ16 3 3 0 0 -1 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ139-ζ133-ζ13 -ζ139+ζ133-ζ13 -ζ1312+ζ1310-ζ134 -ζ1312-ζ1310+ζ134 ζ1311-ζ138-ζ137 -ζ136+ζ135-ζ132 ζ1312-ζ1310-ζ134 ζ136-ζ135-ζ132 -ζ1311-ζ138+ζ137 -ζ139-ζ133+ζ13 -ζ1311+ζ138-ζ137 -ζ136-ζ135+ζ132 complex lifted from C13⋊A4 ρ17 3 3 0 0 -1 0 0 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 -ζ139+ζ133-ζ13 -ζ139-ζ133+ζ13 ζ1312-ζ1310-ζ134 -ζ1312+ζ1310-ζ134 -ζ1311+ζ138-ζ137 ζ136-ζ135-ζ132 -ζ1312-ζ1310+ζ134 -ζ136-ζ135+ζ132 ζ1311-ζ138-ζ137 ζ139-ζ133-ζ13 -ζ1311-ζ138+ζ137 -ζ136+ζ135-ζ132 complex lifted from C13⋊A4 ρ18 3 3 0 0 -1 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 -ζ1312-ζ1310+ζ134 -ζ1312+ζ1310-ζ134 -ζ139+ζ133-ζ13 ζ139-ζ133-ζ13 -ζ136-ζ135+ζ132 -ζ1311+ζ138-ζ137 -ζ139-ζ133+ζ13 -ζ1311-ζ138+ζ137 ζ136-ζ135-ζ132 ζ1312-ζ1310-ζ134 -ζ136+ζ135-ζ132 ζ1311-ζ138-ζ137 complex lifted from C13⋊A4 ρ19 3 3 0 0 3 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ20 3 3 0 0 -1 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 -ζ136-ζ135+ζ132 -ζ136+ζ135-ζ132 -ζ1311+ζ138-ζ137 ζ1311-ζ138-ζ137 -ζ139-ζ133+ζ13 -ζ1312-ζ1310+ζ134 -ζ1311-ζ138+ζ137 -ζ1312+ζ1310-ζ134 -ζ139+ζ133-ζ13 ζ136-ζ135-ζ132 ζ139-ζ133-ζ13 ζ1312-ζ1310-ζ134 complex lifted from C13⋊A4 ρ21 3 3 0 0 -1 0 0 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ136-ζ135-ζ132 -ζ136-ζ135+ζ132 ζ1311-ζ138-ζ137 -ζ1311-ζ138+ζ137 -ζ139+ζ133-ζ13 ζ1312-ζ1310-ζ134 -ζ1311+ζ138-ζ137 -ζ1312-ζ1310+ζ134 ζ139-ζ133-ζ13 -ζ136+ζ135-ζ132 -ζ139-ζ133+ζ13 -ζ1312+ζ1310-ζ134 complex lifted from C13⋊A4 ρ22 3 3 0 0 -1 0 0 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 -ζ1311-ζ138+ζ137 ζ1311-ζ138-ζ137 -ζ136-ζ135+ζ132 ζ136-ζ135-ζ132 -ζ1312+ζ1310-ζ134 -ζ139-ζ133+ζ13 -ζ136+ζ135-ζ132 ζ139-ζ133-ζ13 -ζ1312-ζ1310+ζ134 -ζ1311+ζ138-ζ137 ζ1312-ζ1310-ζ134 -ζ139+ζ133-ζ13 complex lifted from C13⋊A4 ρ23 3 3 0 0 -1 0 0 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 -ζ1312+ζ1310-ζ134 ζ1312-ζ1310-ζ134 -ζ139-ζ133+ζ13 -ζ139+ζ133-ζ13 -ζ136+ζ135-ζ132 -ζ1311-ζ138+ζ137 ζ139-ζ133-ζ13 ζ1311-ζ138-ζ137 -ζ136-ζ135+ζ132 -ζ1312-ζ1310+ζ134 ζ136-ζ135-ζ132 -ζ1311+ζ138-ζ137 complex lifted from C13⋊A4 ρ24 6 -6 0 0 0 0 0 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 -2ζ136-2ζ135-2ζ132 -2ζ1312-2ζ1310-2ζ134 -2ζ139-2ζ133-2ζ13 -2ζ1311-2ζ138-2ζ137 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 6 -6 0 0 0 0 0 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 -2ζ1311-2ζ138-2ζ137 -2ζ139-2ζ133-2ζ13 -2ζ1312-2ζ1310-2ζ134 -2ζ136-2ζ135-2ζ132 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 6 -6 0 0 0 0 0 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 -2ζ139-2ζ133-2ζ13 -2ζ136-2ζ135-2ζ132 -2ζ1311-2ζ138-2ζ137 -2ζ1312-2ζ1310-2ζ134 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ27 6 -6 0 0 0 0 0 2ζ1312+2ζ1310+2ζ134 2ζ1311+2ζ138+2ζ137 2ζ139+2ζ133+2ζ13 2ζ136+2ζ135+2ζ132 -2ζ1312-2ζ1310-2ζ134 -2ζ1311-2ζ138-2ζ137 -2ζ136-2ζ135-2ζ132 -2ζ139-2ζ133-2ζ13 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C26.A4
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 53 14 66)(2 54 15 67)(3 55 16 68)(4 56 17 69)(5 57 18 70)(6 58 19 71)(7 59 20 72)(8 60 21 73)(9 61 22 74)(10 62 23 75)(11 63 24 76)(12 64 25 77)(13 65 26 78)(27 81 40 94)(28 82 41 95)(29 83 42 96)(30 84 43 97)(31 85 44 98)(32 86 45 99)(33 87 46 100)(34 88 47 101)(35 89 48 102)(36 90 49 103)(37 91 50 104)(38 92 51 79)(39 93 52 80)
(1 37 14 50)(2 38 15 51)(3 39 16 52)(4 40 17 27)(5 41 18 28)(6 42 19 29)(7 43 20 30)(8 44 21 31)(9 45 22 32)(10 46 23 33)(11 47 24 34)(12 48 25 35)(13 49 26 36)(53 104 66 91)(54 79 67 92)(55 80 68 93)(56 81 69 94)(57 82 70 95)(58 83 71 96)(59 84 72 97)(60 85 73 98)(61 86 74 99)(62 87 75 100)(63 88 76 101)(64 89 77 102)(65 90 78 103)
(2 4 10)(3 7 19)(5 13 11)(6 16 20)(8 22 12)(9 25 21)(15 17 23)(18 26 24)(27 87 67)(28 90 76)(29 93 59)(30 96 68)(31 99 77)(32 102 60)(33 79 69)(34 82 78)(35 85 61)(36 88 70)(37 91 53)(38 94 62)(39 97 71)(40 100 54)(41 103 63)(42 80 72)(43 83 55)(44 86 64)(45 89 73)(46 92 56)(47 95 65)(48 98 74)(49 101 57)(50 104 66)(51 81 75)(52 84 58)```

`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), (1,53,14,66)(2,54,15,67)(3,55,16,68)(4,56,17,69)(5,57,18,70)(6,58,19,71)(7,59,20,72)(8,60,21,73)(9,61,22,74)(10,62,23,75)(11,63,24,76)(12,64,25,77)(13,65,26,78)(27,81,40,94)(28,82,41,95)(29,83,42,96)(30,84,43,97)(31,85,44,98)(32,86,45,99)(33,87,46,100)(34,88,47,101)(35,89,48,102)(36,90,49,103)(37,91,50,104)(38,92,51,79)(39,93,52,80), (1,37,14,50)(2,38,15,51)(3,39,16,52)(4,40,17,27)(5,41,18,28)(6,42,19,29)(7,43,20,30)(8,44,21,31)(9,45,22,32)(10,46,23,33)(11,47,24,34)(12,48,25,35)(13,49,26,36)(53,104,66,91)(54,79,67,92)(55,80,68,93)(56,81,69,94)(57,82,70,95)(58,83,71,96)(59,84,72,97)(60,85,73,98)(61,86,74,99)(62,87,75,100)(63,88,76,101)(64,89,77,102)(65,90,78,103), (2,4,10)(3,7,19)(5,13,11)(6,16,20)(8,22,12)(9,25,21)(15,17,23)(18,26,24)(27,87,67)(28,90,76)(29,93,59)(30,96,68)(31,99,77)(32,102,60)(33,79,69)(34,82,78)(35,85,61)(36,88,70)(37,91,53)(38,94,62)(39,97,71)(40,100,54)(41,103,63)(42,80,72)(43,83,55)(44,86,64)(45,89,73)(46,92,56)(47,95,65)(48,98,74)(49,101,57)(50,104,66)(51,81,75)(52,84,58)>;`

`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), (1,53,14,66)(2,54,15,67)(3,55,16,68)(4,56,17,69)(5,57,18,70)(6,58,19,71)(7,59,20,72)(8,60,21,73)(9,61,22,74)(10,62,23,75)(11,63,24,76)(12,64,25,77)(13,65,26,78)(27,81,40,94)(28,82,41,95)(29,83,42,96)(30,84,43,97)(31,85,44,98)(32,86,45,99)(33,87,46,100)(34,88,47,101)(35,89,48,102)(36,90,49,103)(37,91,50,104)(38,92,51,79)(39,93,52,80), (1,37,14,50)(2,38,15,51)(3,39,16,52)(4,40,17,27)(5,41,18,28)(6,42,19,29)(7,43,20,30)(8,44,21,31)(9,45,22,32)(10,46,23,33)(11,47,24,34)(12,48,25,35)(13,49,26,36)(53,104,66,91)(54,79,67,92)(55,80,68,93)(56,81,69,94)(57,82,70,95)(58,83,71,96)(59,84,72,97)(60,85,73,98)(61,86,74,99)(62,87,75,100)(63,88,76,101)(64,89,77,102)(65,90,78,103), (2,4,10)(3,7,19)(5,13,11)(6,16,20)(8,22,12)(9,25,21)(15,17,23)(18,26,24)(27,87,67)(28,90,76)(29,93,59)(30,96,68)(31,99,77)(32,102,60)(33,79,69)(34,82,78)(35,85,61)(36,88,70)(37,91,53)(38,94,62)(39,97,71)(40,100,54)(41,103,63)(42,80,72)(43,83,55)(44,86,64)(45,89,73)(46,92,56)(47,95,65)(48,98,74)(49,101,57)(50,104,66)(51,81,75)(52,84,58) );`

`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)], [(1,53,14,66),(2,54,15,67),(3,55,16,68),(4,56,17,69),(5,57,18,70),(6,58,19,71),(7,59,20,72),(8,60,21,73),(9,61,22,74),(10,62,23,75),(11,63,24,76),(12,64,25,77),(13,65,26,78),(27,81,40,94),(28,82,41,95),(29,83,42,96),(30,84,43,97),(31,85,44,98),(32,86,45,99),(33,87,46,100),(34,88,47,101),(35,89,48,102),(36,90,49,103),(37,91,50,104),(38,92,51,79),(39,93,52,80)], [(1,37,14,50),(2,38,15,51),(3,39,16,52),(4,40,17,27),(5,41,18,28),(6,42,19,29),(7,43,20,30),(8,44,21,31),(9,45,22,32),(10,46,23,33),(11,47,24,34),(12,48,25,35),(13,49,26,36),(53,104,66,91),(54,79,67,92),(55,80,68,93),(56,81,69,94),(57,82,70,95),(58,83,71,96),(59,84,72,97),(60,85,73,98),(61,86,74,99),(62,87,75,100),(63,88,76,101),(64,89,77,102),(65,90,78,103)], [(2,4,10),(3,7,19),(5,13,11),(6,16,20),(8,22,12),(9,25,21),(15,17,23),(18,26,24),(27,87,67),(28,90,76),(29,93,59),(30,96,68),(31,99,77),(32,102,60),(33,79,69),(34,82,78),(35,85,61),(36,88,70),(37,91,53),(38,94,62),(39,97,71),(40,100,54),(41,103,63),(42,80,72),(43,83,55),(44,86,64),(45,89,73),(46,92,56),(47,95,65),(48,98,74),(49,101,57),(50,104,66),(51,81,75),(52,84,58)]])`

Matrix representation of C26.A4 in GL5(𝔽157)

 156 0 0 0 0 0 156 0 0 0 0 0 134 144 69 0 0 69 82 110 0 0 110 134 46
,
 132 145 0 0 0 26 25 0 0 0 0 0 71 40 139 0 0 139 146 83 0 0 83 81 96
,
 1 1 0 0 0 155 156 0 0 0 0 0 29 156 144 0 0 144 57 126 0 0 126 90 70
,
 1 1 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 110 134 46 0 0 97 47 22

`G:=sub<GL(5,GF(157))| [156,0,0,0,0,0,156,0,0,0,0,0,134,69,110,0,0,144,82,134,0,0,69,110,46],[132,26,0,0,0,145,25,0,0,0,0,0,71,139,83,0,0,40,146,81,0,0,139,83,96],[1,155,0,0,0,1,156,0,0,0,0,0,29,144,126,0,0,156,57,90,0,0,144,126,70],[1,0,0,0,0,1,12,0,0,0,0,0,1,110,97,0,0,0,134,47,0,0,0,46,22] >;`

C26.A4 in GAP, Magma, Sage, TeX

`C_{26}.A_4`
`% in TeX`

`G:=Group("C26.A4");`
`// GroupNames label`

`G:=SmallGroup(312,26);`
`// by ID`

`G=gap.SmallGroup(312,26);`
`# by ID`

`G:=PCGroup([5,-3,-2,2,-13,-2,61,526,137,817,402,723]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^26=d^3=1,b^2=c^2=a^13,a*b=b*a,a*c=c*a,d*a*d^-1=a^9,c*b*c^-1=a^13*b,d*b*d^-1=a^13*b*c,d*c*d^-1=b>;`
`// generators/relations`

Export

׿
×
𝔽