direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4xC31, C4:C62, C22:C62, C124:3C2, C62.6C22, (C2xC62):1C2, C2.1(C2xC62), SmallGroup(248,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4xC31
G = < a,b,c | a31=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C22, D4, C31, C62, C2xC62, D4xC31
Smallest permutation representation of D4xC31
►On 124 points
Generators in S124
(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 66 35 99)(2 67 36 100)(3 68 37 101)(4 69 38 102)(5 70 39 103)(6 71 40 104)(7 72 41 105)(8 73 42 106)(9 74 43 107)(10 75 44 108)(11 76 45 109)(12 77 46 110)(13 78 47 111)(14 79 48 112)(15 80 49 113)(16 81 50 114)(17 82 51 115)(18 83 52 116)(19 84 53 117)(20 85 54 118)(21 86 55 119)(22 87 56 120)(23 88 57 121)(24 89 58 122)(25 90 59 123)(26 91 60 124)(27 92 61 94)(28 93 62 95)(29 63 32 96)(30 64 33 97)(31 65 34 98)
(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 124)(92 94)(93 95)
G:=sub<Sym(124)| (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), (1,66,35,99)(2,67,36,100)(3,68,37,101)(4,69,38,102)(5,70,39,103)(6,71,40,104)(7,72,41,105)(8,73,42,106)(9,74,43,107)(10,75,44,108)(11,76,45,109)(12,77,46,110)(13,78,47,111)(14,79,48,112)(15,80,49,113)(16,81,50,114)(17,82,51,115)(18,83,52,116)(19,84,53,117)(20,85,54,118)(21,86,55,119)(22,87,56,120)(23,88,57,121)(24,89,58,122)(25,90,59,123)(26,91,60,124)(27,92,61,94)(28,93,62,95)(29,63,32,96)(30,64,33,97)(31,65,34,98), (63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,94)(93,95)>;
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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,66,35,99)(2,67,36,100)(3,68,37,101)(4,69,38,102)(5,70,39,103)(6,71,40,104)(7,72,41,105)(8,73,42,106)(9,74,43,107)(10,75,44,108)(11,76,45,109)(12,77,46,110)(13,78,47,111)(14,79,48,112)(15,80,49,113)(16,81,50,114)(17,82,51,115)(18,83,52,116)(19,84,53,117)(20,85,54,118)(21,86,55,119)(22,87,56,120)(23,88,57,121)(24,89,58,122)(25,90,59,123)(26,91,60,124)(27,92,61,94)(28,93,62,95)(29,63,32,96)(30,64,33,97)(31,65,34,98), (63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,94)(93,95) );
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),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,66,35,99),(2,67,36,100),(3,68,37,101),(4,69,38,102),(5,70,39,103),(6,71,40,104),(7,72,41,105),(8,73,42,106),(9,74,43,107),(10,75,44,108),(11,76,45,109),(12,77,46,110),(13,78,47,111),(14,79,48,112),(15,80,49,113),(16,81,50,114),(17,82,51,115),(18,83,52,116),(19,84,53,117),(20,85,54,118),(21,86,55,119),(22,87,56,120),(23,88,57,121),(24,89,58,122),(25,90,59,123),(26,91,60,124),(27,92,61,94),(28,93,62,95),(29,63,32,96),(30,64,33,97),(31,65,34,98)], [(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(79,112),(80,113),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,121),(89,122),(90,123),(91,124),(92,94),(93,95)]])
D4xC31 is a maximal subgroup of
D4:D31 D4.D31 D4:2D31
155 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 31A | ··· | 31AD | 62A | ··· | 62AD | 62AE | ··· | 62CL | 124A | ··· | 124AD |
| order | 1 | 2 | 2 | 2 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
155 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C31 | C62 | C62 | D4 | D4xC31 |
| kernel | D4xC31 | C124 | C2xC62 | D4 | C4 | C22 | C31 | C1 |
| # reps | 1 | 1 | 2 | 30 | 30 | 60 | 1 | 30 |
Matrix representation of D4xC31 ►in GL2(F373) generated by
| 215 | 0 |
| 0 | 215 |
| 328 | 267 |
| 371 | 45 |
| 1 | 328 |
| 0 | 372 |
G:=sub<GL(2,GF(373))| [215,0,0,215],[328,371,267,45],[1,0,328,372] >;
D4xC31 in GAP, Magma, Sage, TeX
D_4\times C_{31} % in TeX
G:=Group("D4xC31"); // GroupNames label
G:=SmallGroup(248,9);
// by ID
G=gap.SmallGroup(248,9);
# by ID
G:=PCGroup([4,-2,-2,-31,-2,1009]);
// Polycyclic
G:=Group<a,b,c|a^31=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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