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## G = A4×C19order 228 = 22·3·19

### Direct product of C19 and A4

Aliases: A4×C19, C22⋊C57, (C2×C38)⋊1C3, SmallGroup(228,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C19
 Chief series C1 — C22 — C2×C38 — A4×C19
 Lower central C22 — A4×C19
 Upper central C1 — C19

Generators and relations for A4×C19
G = < a,b,c,d | a19=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C19
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 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 54)(21 55)(22 56)(23 57)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)
(1 57)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 58)(36 59)(37 60)(38 61)
(20 62 54)(21 63 55)(22 64 56)(23 65 57)(24 66 39)(25 67 40)(26 68 41)(27 69 42)(28 70 43)(29 71 44)(30 72 45)(31 73 46)(32 74 47)(33 75 48)(34 76 49)(35 58 50)(36 59 51)(37 60 52)(38 61 53)

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

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

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

A4×C19 is a maximal subgroup of   C19⋊S4

76 conjugacy classes

 class 1 2 3A 3B 19A ··· 19R 38A ··· 38R 57A ··· 57AJ order 1 2 3 3 19 ··· 19 38 ··· 38 57 ··· 57 size 1 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4

76 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C3 C19 C57 A4 A4×C19 kernel A4×C19 C2×C38 A4 C22 C19 C1 # reps 1 2 18 36 1 18

Matrix representation of A4×C19 in GL3(𝔽229) generated by

 225 0 0 0 225 0 0 0 225
,
 228 228 228 0 0 1 0 1 0
,
 0 1 0 1 0 0 228 228 228
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(229))| [225,0,0,0,225,0,0,0,225],[228,0,0,228,0,1,228,1,0],[0,1,228,1,0,228,0,0,228],[0,0,1,1,0,0,0,1,0] >;

A4×C19 in GAP, Magma, Sage, TeX

A_4\times C_{19}
% in TeX

G:=Group("A4xC19");
// GroupNames label

G:=SmallGroup(228,10);
// by ID

G=gap.SmallGroup(228,10);
# by ID

G:=PCGroup([4,-3,-19,-2,2,1370,2739]);
// Polycyclic

G:=Group<a,b,c,d|a^19=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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