Copied to
clipboard

## G = C17⋊S4order 408 = 23·3·17

### The semidirect product of C17 and S4 acting via S4/A4=C2

Aliases: C17⋊S4, A4⋊D17, C22⋊D51, (C2×C34)⋊2S3, (A4×C17)⋊1C2, SmallGroup(408,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C17 — C17⋊S4
 Chief series C1 — C22 — C2×C34 — A4×C17 — C17⋊S4
 Lower central A4×C17 — C17⋊S4
 Upper central C1

Generators and relations for C17⋊S4
G = < a,b,c,d,e | a17=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
102C2
4C3
51C22
51C4
68S3
3C34
6D17
4C51
51D4
3D34
4D51
17S4

Smallest permutation representation of C17⋊S4
On 68 points
Generators in S68
```(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)
(1 33)(2 34)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(35 67)(36 68)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)
(1 66)(2 67)(3 68)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 35)
(18 68 36)(19 52 37)(20 53 38)(21 54 39)(22 55 40)(23 56 41)(24 57 42)(25 58 43)(26 59 44)(27 60 45)(28 61 46)(29 62 47)(30 63 48)(31 64 49)(32 65 50)(33 66 51)(34 67 35)
(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 51)(34 50)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(64 68)(65 67)```

`G:=sub<Sym(68)| (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), (1,33)(2,34)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(35,67)(36,68)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66), (1,66)(2,67)(3,68)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,35), (18,68,36)(19,52,37)(20,53,38)(21,54,39)(22,55,40)(23,56,41)(24,57,42)(25,58,43)(26,59,44)(27,60,45)(28,61,46)(29,62,47)(30,63,48)(31,64,49)(32,65,50)(33,66,51)(34,67,35), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,51)(34,50)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)>;`

`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), (1,33)(2,34)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(35,67)(36,68)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66), (1,66)(2,67)(3,68)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,35), (18,68,36)(19,52,37)(20,53,38)(21,54,39)(22,55,40)(23,56,41)(24,57,42)(25,58,43)(26,59,44)(27,60,45)(28,61,46)(29,62,47)(30,63,48)(31,64,49)(32,65,50)(33,66,51)(34,67,35), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,51)(34,50)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67) );`

`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)], [(1,33),(2,34),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(35,67),(36,68),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66)], [(1,66),(2,67),(3,68),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,35)], [(18,68,36),(19,52,37),(20,53,38),(21,54,39),(22,55,40),(23,56,41),(24,57,42),(25,58,43),(26,59,44),(27,60,45),(28,61,46),(29,62,47),(30,63,48),(31,64,49),(32,65,50),(33,66,51),(34,67,35)], [(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,51),(34,50),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(64,68),(65,67)]])`

37 conjugacy classes

 class 1 2A 2B 3 4 17A ··· 17H 34A ··· 34H 51A ··· 51P order 1 2 2 3 4 17 ··· 17 34 ··· 34 51 ··· 51 size 1 3 102 8 102 2 ··· 2 6 ··· 6 8 ··· 8

37 irreducible representations

 dim 1 1 2 2 2 3 6 type + + + + + + + image C1 C2 S3 D17 D51 S4 C17⋊S4 kernel C17⋊S4 A4×C17 C2×C34 A4 C22 C17 C1 # reps 1 1 1 8 16 2 8

Matrix representation of C17⋊S4 in GL5(𝔽409)

 53 46 0 0 0 363 99 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 408 408 408 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 408 408 408 0 0 0 0 1 0 0 0 1 0
,
 0 408 0 0 0 1 408 0 0 0 0 0 1 0 0 0 0 408 408 408 0 0 0 1 0
,
 408 0 0 0 0 408 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(409))| [53,363,0,0,0,46,99,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,408,1,0,0,0,408,0,0,0,1,408,0],[1,0,0,0,0,0,1,0,0,0,0,0,408,0,0,0,0,408,0,1,0,0,408,1,0],[0,1,0,0,0,408,408,0,0,0,0,0,1,408,0,0,0,0,408,1,0,0,0,408,0],[408,408,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C17⋊S4 in GAP, Magma, Sage, TeX

`C_{17}\rtimes S_4`
`% in TeX`

`G:=Group("C17:S4");`
`// GroupNames label`

`G:=SmallGroup(408,37);`
`// by ID`

`G=gap.SmallGroup(408,37);`
`# by ID`

`G:=PCGroup([5,-2,-3,-17,-2,2,41,1442,4083,2048,2554,3834]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^17=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

Export

׿
×
𝔽