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## G = C17×S4order 408 = 23·3·17

### Direct product of C17 and S4

Aliases: C17×S4, A4⋊C34, (C2×C34)⋊1S3, C22⋊(S3×C17), (A4×C17)⋊3C2, SmallGroup(408,36)

Series: Derived Chief Lower central Upper central

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

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

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

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

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

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

85 conjugacy classes

 class 1 2A 2B 3 4 17A ··· 17P 34A ··· 34P 34Q ··· 34AF 51A ··· 51P 68A ··· 68P order 1 2 2 3 4 17 ··· 17 34 ··· 34 34 ··· 34 51 ··· 51 68 ··· 68 size 1 3 6 8 6 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6

85 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + + + image C1 C2 C17 C34 S3 S3×C17 S4 C17×S4 kernel C17×S4 A4×C17 S4 A4 C2×C34 C22 C17 C1 # reps 1 1 16 16 1 16 2 32

Matrix representation of C17×S4 in GL3(𝔽409) generated by

 180 0 0 0 180 0 0 0 180
,
 0 408 1 0 408 0 1 408 0
,
 408 0 0 408 0 1 408 1 0
,
 1 0 408 0 0 408 0 1 408
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(409))| [180,0,0,0,180,0,0,0,180],[0,0,1,408,408,408,1,0,0],[408,408,408,0,0,1,0,1,0],[1,0,0,0,0,1,408,408,408],[1,0,0,0,0,1,0,1,0] >;

C17×S4 in GAP, Magma, Sage, TeX

C_{17}\times S_4
% in TeX

G:=Group("C17xS4");
// GroupNames label

G:=SmallGroup(408,36);
// by ID

G=gap.SmallGroup(408,36);
# by ID

G:=PCGroup([5,-2,-17,-3,-2,2,1022,4083,133,2554,239]);
// 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,a*e=e*a,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

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