direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C38, C23⋊C57, C22⋊C114, (C2×C38)⋊6C6, (C22×C38)⋊1C3, SmallGroup(456,49)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×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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