direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D31, C4⋊1D62, C124⋊C22, D124⋊3C2, C22⋊1D62, D62⋊2C22, C62.5C23, Dic31⋊1C22, C31⋊2(C2×D4), (C2×C62)⋊C22, (C4×D31)⋊1C2, (D4×C31)⋊2C2, C31⋊D4⋊1C2, (C22×D31)⋊2C2, C2.6(C22×D31), SmallGroup(496,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D31
G = < a,b,c,d | a4=b2=c31=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 820 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C31, D31, D31, C62, C62, Dic31, C124, D62, D62, D62, C2×C62, C4×D31, D124, C31⋊D4, D4×C31, C22×D31, D4×D31
Quotients: C1, C2, C22, D4, C23, C2×D4, D31, D62, C22×D31, D4×D31
►On 124 points
Generators in S124
(1 106 46 68)(2 107 47 69)(3 108 48 70)(4 109 49 71)(5 110 50 72)(6 111 51 73)(7 112 52 74)(8 113 53 75)(9 114 54 76)(10 115 55 77)(11 116 56 78)(12 117 57 79)(13 118 58 80)(14 119 59 81)(15 120 60 82)(16 121 61 83)(17 122 62 84)(18 123 32 85)(19 124 33 86)(20 94 34 87)(21 95 35 88)(22 96 36 89)(23 97 37 90)(24 98 38 91)(25 99 39 92)(26 100 40 93)(27 101 41 63)(28 102 42 64)(29 103 43 65)(30 104 44 66)(31 105 45 67)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 63)(28 64)(29 65)(30 66)(31 67)(32 123)(33 124)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(60 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(118 124)(119 123)(120 122)
G:=sub<Sym(124)| (1,106,46,68)(2,107,47,69)(3,108,48,70)(4,109,49,71)(5,110,50,72)(6,111,51,73)(7,112,52,74)(8,113,53,75)(9,114,54,76)(10,115,55,77)(11,116,56,78)(12,117,57,79)(13,118,58,80)(14,119,59,81)(15,120,60,82)(16,121,61,83)(17,122,62,84)(18,123,32,85)(19,124,33,86)(20,94,34,87)(21,95,35,88)(22,96,36,89)(23,97,37,90)(24,98,38,91)(25,99,39,92)(26,100,40,93)(27,101,41,63)(28,102,42,64)(29,103,43,65)(30,104,44,66)(31,105,45,67), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,63)(28,64)(29,65)(30,66)(31,67)(32,123)(33,124)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122), (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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(60,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(118,124)(119,123)(120,122)>;
G:=Group( (1,106,46,68)(2,107,47,69)(3,108,48,70)(4,109,49,71)(5,110,50,72)(6,111,51,73)(7,112,52,74)(8,113,53,75)(9,114,54,76)(10,115,55,77)(11,116,56,78)(12,117,57,79)(13,118,58,80)(14,119,59,81)(15,120,60,82)(16,121,61,83)(17,122,62,84)(18,123,32,85)(19,124,33,86)(20,94,34,87)(21,95,35,88)(22,96,36,89)(23,97,37,90)(24,98,38,91)(25,99,39,92)(26,100,40,93)(27,101,41,63)(28,102,42,64)(29,103,43,65)(30,104,44,66)(31,105,45,67), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,63)(28,64)(29,65)(30,66)(31,67)(32,123)(33,124)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122), (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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(60,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(118,124)(119,123)(120,122) );
G=PermutationGroup([[(1,106,46,68),(2,107,47,69),(3,108,48,70),(4,109,49,71),(5,110,50,72),(6,111,51,73),(7,112,52,74),(8,113,53,75),(9,114,54,76),(10,115,55,77),(11,116,56,78),(12,117,57,79),(13,118,58,80),(14,119,59,81),(15,120,60,82),(16,121,61,83),(17,122,62,84),(18,123,32,85),(19,124,33,86),(20,94,34,87),(21,95,35,88),(22,96,36,89),(23,97,37,90),(24,98,38,91),(25,99,39,92),(26,100,40,93),(27,101,41,63),(28,102,42,64),(29,103,43,65),(30,104,44,66),(31,105,45,67)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,63),(28,64),(29,65),(30,66),(31,67),(32,123),(33,124),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122)], [(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),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(60,62),(63,72),(64,71),(65,70),(66,69),(67,68),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(118,124),(119,123),(120,122)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 31A | ··· | 31O | 62A | ··· | 62O | 62P | ··· | 62AS | 124A | ··· | 124O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 2 | 2 | 31 | 31 | 62 | 62 | 2 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D31 | D62 | D62 | D4×D31 |
| kernel | D4×D31 | C4×D31 | D124 | C31⋊D4 | D4×C31 | C22×D31 | D31 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 15 | 15 | 30 | 15 |
Matrix representation of D4×D31 ►in GL4(𝔽373) generated by
| 372 | 0 | 0 | 0 |
| 0 | 372 | 0 | 0 |
| 0 | 0 | 32 | 356 |
| 0 | 0 | 170 | 341 |
| 372 | 0 | 0 | 0 |
| 0 | 372 | 0 | 0 |
| 0 | 0 | 32 | 356 |
| 0 | 0 | 126 | 341 |
| 350 | 1 | 0 | 0 |
| 39 | 128 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 346 | 369 | 0 | 0 |
| 182 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(373))| [372,0,0,0,0,372,0,0,0,0,32,170,0,0,356,341],[372,0,0,0,0,372,0,0,0,0,32,126,0,0,356,341],[350,39,0,0,1,128,0,0,0,0,1,0,0,0,0,1],[346,182,0,0,369,27,0,0,0,0,1,0,0,0,0,1] >;
D4×D31 in GAP, Magma, Sage, TeX
D_4\times D_{31} % in TeX
G:=Group("D4xD31"); // GroupNames label
G:=SmallGroup(496,31);
// by ID
G=gap.SmallGroup(496,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,97,12004]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^31=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations