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## G = A4×C38order 456 = 23·3·19

### Direct product of C38 and A4

Aliases: A4×C38, C23⋊C57, C22⋊C114, (C2×C38)⋊6C6, (C22×C38)⋊1C3, SmallGroup(456,49)

Series: Derived Chief Lower central Upper central

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

Generators and relations for A4×C38
G = < a,b,c,d | a38=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C38
On 114 points
Generators in S114
(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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)
(1 97 63)(2 98 64)(3 99 65)(4 100 66)(5 101 67)(6 102 68)(7 103 69)(8 104 70)(9 105 71)(10 106 72)(11 107 73)(12 108 74)(13 109 75)(14 110 76)(15 111 39)(16 112 40)(17 113 41)(18 114 42)(19 77 43)(20 78 44)(21 79 45)(22 80 46)(23 81 47)(24 82 48)(25 83 49)(26 84 50)(27 85 51)(28 86 52)(29 87 53)(30 88 54)(31 89 55)(32 90 56)(33 91 57)(34 92 58)(35 93 59)(36 94 60)(37 95 61)(38 96 62)

G:=sub<Sym(114)| (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114), (1,97,63)(2,98,64)(3,99,65)(4,100,66)(5,101,67)(6,102,68)(7,103,69)(8,104,70)(9,105,71)(10,106,72)(11,107,73)(12,108,74)(13,109,75)(14,110,76)(15,111,39)(16,112,40)(17,113,41)(18,114,42)(19,77,43)(20,78,44)(21,79,45)(22,80,46)(23,81,47)(24,82,48)(25,83,49)(26,84,50)(27,85,51)(28,86,52)(29,87,53)(30,88,54)(31,89,55)(32,90,56)(33,91,57)(34,92,58)(35,93,59)(36,94,60)(37,95,61)(38,96,62)>;

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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114), (1,97,63)(2,98,64)(3,99,65)(4,100,66)(5,101,67)(6,102,68)(7,103,69)(8,104,70)(9,105,71)(10,106,72)(11,107,73)(12,108,74)(13,109,75)(14,110,76)(15,111,39)(16,112,40)(17,113,41)(18,114,42)(19,77,43)(20,78,44)(21,79,45)(22,80,46)(23,81,47)(24,82,48)(25,83,49)(26,84,50)(27,85,51)(28,86,52)(29,87,53)(30,88,54)(31,89,55)(32,90,56)(33,91,57)(34,92,58)(35,93,59)(36,94,60)(37,95,61)(38,96,62) );

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),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(77,96),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114)], [(1,97,63),(2,98,64),(3,99,65),(4,100,66),(5,101,67),(6,102,68),(7,103,69),(8,104,70),(9,105,71),(10,106,72),(11,107,73),(12,108,74),(13,109,75),(14,110,76),(15,111,39),(16,112,40),(17,113,41),(18,114,42),(19,77,43),(20,78,44),(21,79,45),(22,80,46),(23,81,47),(24,82,48),(25,83,49),(26,84,50),(27,85,51),(28,86,52),(29,87,53),(30,88,54),(31,89,55),(32,90,56),(33,91,57),(34,92,58),(35,93,59),(36,94,60),(37,95,61),(38,96,62)]])

152 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 19A ··· 19R 38A ··· 38R 38S ··· 38BB 57A ··· 57AJ 114A ··· 114AJ order 1 2 2 2 3 3 6 6 19 ··· 19 38 ··· 38 38 ··· 38 57 ··· 57 114 ··· 114 size 1 1 3 3 4 4 4 4 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

152 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 C19 C38 C57 C114 A4 C2×A4 A4×C19 A4×C38 kernel A4×C38 A4×C19 C22×C38 C2×C38 C2×A4 A4 C23 C22 C38 C19 C2 C1 # reps 1 1 2 2 18 18 36 36 1 1 18 18

Matrix representation of A4×C38 in GL3(𝔽229) generated by

 172 0 0 0 172 0 0 0 172
,
 228 0 0 0 228 0 135 0 1
,
 228 0 0 95 1 0 0 0 228
,
 134 227 0 0 95 1 0 135 0
G:=sub<GL(3,GF(229))| [172,0,0,0,172,0,0,0,172],[228,0,135,0,228,0,0,0,1],[228,95,0,0,1,0,0,0,228],[134,0,0,227,95,135,0,1,0] >;

A4×C38 in GAP, Magma, Sage, TeX

A_4\times C_{38}
% in TeX

G:=Group("A4xC38");
// GroupNames label

G:=SmallGroup(456,49);
// by ID

G=gap.SmallGroup(456,49);
# by ID

G:=PCGroup([5,-2,-3,-19,-2,2,2288,4284]);
// Polycyclic

G:=Group<a,b,c,d|a^38=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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