metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D123, C41⋊S3, C3⋊D41, C123⋊1C2, sometimes denoted D246 or Dih123 or Dih246, SmallGroup(246,3)
Series: Derived ►Chief ►Lower central ►Upper central
C123 — D123 |
Generators and relations for D123
G = < a,b | a123=b2=1, bab=a-1 >
(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)
(1 123)(2 122)(3 121)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)
G:=sub<Sym(123)| (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), (1,123)(2,122)(3,121)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)>;
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), (1,123)(2,122)(3,121)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,103)(22,102)(23,101)(24,100)(25,99)(26,98)(27,97)(28,96)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63) );
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)], [(1,123),(2,122),(3,121),(4,120),(5,119),(6,118),(7,117),(8,116),(9,115),(10,114),(11,113),(12,112),(13,111),(14,110),(15,109),(16,108),(17,107),(18,106),(19,105),(20,104),(21,103),(22,102),(23,101),(24,100),(25,99),(26,98),(27,97),(28,96),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63)]])
D123 is a maximal subgroup of
S3×D41
D123 is a maximal quotient of Dic123
63 conjugacy classes
class | 1 | 2 | 3 | 41A | ··· | 41T | 123A | ··· | 123AN |
order | 1 | 2 | 3 | 41 | ··· | 41 | 123 | ··· | 123 |
size | 1 | 123 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
dim | 1 | 1 | 2 | 2 | 2 |
type | + | + | + | + | + |
image | C1 | C2 | S3 | D41 | D123 |
kernel | D123 | C123 | C41 | C3 | C1 |
# reps | 1 | 1 | 1 | 20 | 40 |
Matrix representation of D123 ►in GL2(𝔽739) generated by
135 | 443 |
296 | 90 |
135 | 443 |
481 | 604 |
G:=sub<GL(2,GF(739))| [135,296,443,90],[135,481,443,604] >;
D123 in GAP, Magma, Sage, TeX
D_{123}
% in TeX
G:=Group("D123");
// GroupNames label
G:=SmallGroup(246,3);
// by ID
G=gap.SmallGroup(246,3);
# by ID
G:=PCGroup([3,-2,-3,-41,25,2162]);
// Polycyclic
G:=Group<a,b|a^123=b^2=1,b*a*b=a^-1>;
// generators/relations
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