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## G = C97⋊C3order 291 = 3·97

### The semidirect product of C97 and C3 acting faithfully

Aliases: C97⋊C3, SmallGroup(291,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C97 — C97⋊C3
 Chief series C1 — C97 — C97⋊C3
 Lower central C97 — C97⋊C3
 Upper central C1

Generators and relations for C97⋊C3
G = < a,b | a97=b3=1, bab-1=a61 >

Smallest permutation representation of C97⋊C3
On 97 points: primitive
Generators in S97
```(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)
(2 36 62)(3 71 26)(4 9 87)(5 44 51)(6 79 15)(7 17 76)(8 52 40)(10 25 65)(11 60 29)(12 95 90)(13 33 54)(14 68 18)(16 41 43)(19 49 32)(20 84 93)(21 22 57)(23 92 82)(24 30 46)(27 38 35)(28 73 96)(31 81 85)(34 89 74)(37 97 63)(39 70 88)(42 78 77)(45 86 66)(47 59 91)(48 94 55)(50 67 80)(53 75 69)(56 83 58)(61 64 72)```

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

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

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

35 conjugacy classes

 class 1 3A 3B 97A ··· 97AF order 1 3 3 97 ··· 97 size 1 97 97 3 ··· 3

35 irreducible representations

 dim 1 1 3 type + image C1 C3 C97⋊C3 kernel C97⋊C3 C97 C1 # reps 1 2 32

Matrix representation of C97⋊C3 in GL3(𝔽1747) generated by

 894 1 0 1549 0 1 1196 180 392
,
 46 515 945 7 163 41 92 957 1538
`G:=sub<GL(3,GF(1747))| [894,1549,1196,1,0,180,0,1,392],[46,7,92,515,163,957,945,41,1538] >;`

C97⋊C3 in GAP, Magma, Sage, TeX

`C_{97}\rtimes C_3`
`% in TeX`

`G:=Group("C97:C3");`
`// GroupNames label`

`G:=SmallGroup(291,1);`
`// by ID`

`G=gap.SmallGroup(291,1);`
`# by ID`

`G:=PCGroup([2,-3,-97,421]);`
`// Polycyclic`

`G:=Group<a,b|a^97=b^3=1,b*a*b^-1=a^61>;`
`// generators/relations`

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