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G = C19×S4order 456 = 23·3·19

Direct product of C19 and S4

Aliases: C19×S4, A4⋊C38, (C2×C38)⋊1S3, C22⋊(S3×C19), (A4×C19)⋊3C2, SmallGroup(456,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C19×S4
 Chief series C1 — C22 — A4 — A4×C19 — C19×S4
 Lower central A4 — C19×S4
 Upper central C1 — C19

Generators and relations for C19×S4
G = < a,b,c,d,e | a19=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 C19×S4
On 76 points
Generators in S76
(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)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)
(1 38)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(39 73)(40 74)(41 75)(42 76)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)
(20 39 73)(21 40 74)(22 41 75)(23 42 76)(24 43 58)(25 44 59)(26 45 60)(27 46 61)(28 47 62)(29 48 63)(30 49 64)(31 50 65)(32 51 66)(33 52 67)(34 53 68)(35 54 69)(36 55 70)(37 56 71)(38 57 72)
(39 73)(40 74)(41 75)(42 76)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)

G:=sub<Sym(76)| (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), (1,72)(2,73)(3,74)(4,75)(5,76)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57), (1,38)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(39,73)(40,74)(41,75)(42,76)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72), (20,39,73)(21,40,74)(22,41,75)(23,42,76)(24,43,58)(25,44,59)(26,45,60)(27,46,61)(28,47,62)(29,48,63)(30,49,64)(31,50,65)(32,51,66)(33,52,67)(34,53,68)(35,54,69)(36,55,70)(37,56,71)(38,57,72), (39,73)(40,74)(41,75)(42,76)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)>;

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), (1,72)(2,73)(3,74)(4,75)(5,76)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57), (1,38)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(39,73)(40,74)(41,75)(42,76)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72), (20,39,73)(21,40,74)(22,41,75)(23,42,76)(24,43,58)(25,44,59)(26,45,60)(27,46,61)(28,47,62)(29,48,63)(30,49,64)(31,50,65)(32,51,66)(33,52,67)(34,53,68)(35,54,69)(36,55,70)(37,56,71)(38,57,72), (39,73)(40,74)(41,75)(42,76)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72) );

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)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57)], [(1,38),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(39,73),(40,74),(41,75),(42,76),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72)], [(20,39,73),(21,40,74),(22,41,75),(23,42,76),(24,43,58),(25,44,59),(26,45,60),(27,46,61),(28,47,62),(29,48,63),(30,49,64),(31,50,65),(32,51,66),(33,52,67),(34,53,68),(35,54,69),(36,55,70),(37,56,71),(38,57,72)], [(39,73),(40,74),(41,75),(42,76),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72)]])

95 conjugacy classes

 class 1 2A 2B 3 4 19A ··· 19R 38A ··· 38R 38S ··· 38AJ 57A ··· 57R 76A ··· 76R order 1 2 2 3 4 19 ··· 19 38 ··· 38 38 ··· 38 57 ··· 57 76 ··· 76 size 1 3 6 8 6 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6

95 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + + + image C1 C2 C19 C38 S3 S3×C19 S4 C19×S4 kernel C19×S4 A4×C19 S4 A4 C2×C38 C22 C19 C1 # reps 1 1 18 18 1 18 2 36

Matrix representation of C19×S4 in GL3(𝔽229) generated by

 53 0 0 0 53 0 0 0 53
,
 0 0 1 228 228 228 1 0 0
,
 228 228 228 0 0 1 0 1 0
,
 1 0 0 228 228 228 0 1 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(229))| [53,0,0,0,53,0,0,0,53],[0,228,1,0,228,0,1,228,0],[228,0,0,228,0,1,228,1,0],[1,228,0,0,228,1,0,228,0],[1,0,0,0,0,1,0,1,0] >;

C19×S4 in GAP, Magma, Sage, TeX

C_{19}\times S_4
% in TeX

G:=Group("C19xS4");
// GroupNames label

G:=SmallGroup(456,42);
// by ID

G=gap.SmallGroup(456,42);
# by ID

G:=PCGroup([5,-2,-19,-3,-2,2,1142,4563,133,2854,239]);
// Polycyclic

G:=Group<a,b,c,d,e|a^19=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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