metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D33⋊C4, D66.C2, C6.3D22, C22.3D6, Dic11⋊2S3, Dic3⋊2D11, C66.3C22, C11⋊1(C4×S3), C33⋊3(C2×C4), C3⋊1(C4×D11), C2.3(S3×D11), (C3×Dic11)⋊2C2, (C11×Dic3)⋊2C2, SmallGroup(264,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — D33⋊C4 |
Generators and relations for D33⋊C4
G = < a,b,c | a33=b2=c4=1, bab=a-1, cac-1=a23, cbc-1=a22b >
Smallest permutation representation of D33⋊C4
►On 132 points
Generators in S132
(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)(100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(100 125)(101 124)(102 123)(103 122)(104 121)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(126 132)(127 131)(128 130)
(1 113 58 75)(2 103 59 98)(3 126 60 88)(4 116 61 78)(5 106 62 68)(6 129 63 91)(7 119 64 81)(8 109 65 71)(9 132 66 94)(10 122 34 84)(11 112 35 74)(12 102 36 97)(13 125 37 87)(14 115 38 77)(15 105 39 67)(16 128 40 90)(17 118 41 80)(18 108 42 70)(19 131 43 93)(20 121 44 83)(21 111 45 73)(22 101 46 96)(23 124 47 86)(24 114 48 76)(25 104 49 99)(26 127 50 89)(27 117 51 79)(28 107 52 69)(29 130 53 92)(30 120 54 82)(31 110 55 72)(32 100 56 95)(33 123 57 85)
G:=sub<Sym(132)| (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)(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(126,132)(127,131)(128,130), (1,113,58,75)(2,103,59,98)(3,126,60,88)(4,116,61,78)(5,106,62,68)(6,129,63,91)(7,119,64,81)(8,109,65,71)(9,132,66,94)(10,122,34,84)(11,112,35,74)(12,102,36,97)(13,125,37,87)(14,115,38,77)(15,105,39,67)(16,128,40,90)(17,118,41,80)(18,108,42,70)(19,131,43,93)(20,121,44,83)(21,111,45,73)(22,101,46,96)(23,124,47,86)(24,114,48,76)(25,104,49,99)(26,127,50,89)(27,117,51,79)(28,107,52,69)(29,130,53,92)(30,120,54,82)(31,110,55,72)(32,100,56,95)(33,123,57,85)>;
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)(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)(126,132)(127,131)(128,130), (1,113,58,75)(2,103,59,98)(3,126,60,88)(4,116,61,78)(5,106,62,68)(6,129,63,91)(7,119,64,81)(8,109,65,71)(9,132,66,94)(10,122,34,84)(11,112,35,74)(12,102,36,97)(13,125,37,87)(14,115,38,77)(15,105,39,67)(16,128,40,90)(17,118,41,80)(18,108,42,70)(19,131,43,93)(20,121,44,83)(21,111,45,73)(22,101,46,96)(23,124,47,86)(24,114,48,76)(25,104,49,99)(26,127,50,89)(27,117,51,79)(28,107,52,69)(29,130,53,92)(30,120,54,82)(31,110,55,72)(32,100,56,95)(33,123,57,85) );
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),(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(49,66),(50,65),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(100,125),(101,124),(102,123),(103,122),(104,121),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113),(126,132),(127,131),(128,130)], [(1,113,58,75),(2,103,59,98),(3,126,60,88),(4,116,61,78),(5,106,62,68),(6,129,63,91),(7,119,64,81),(8,109,65,71),(9,132,66,94),(10,122,34,84),(11,112,35,74),(12,102,36,97),(13,125,37,87),(14,115,38,77),(15,105,39,67),(16,128,40,90),(17,118,41,80),(18,108,42,70),(19,131,43,93),(20,121,44,83),(21,111,45,73),(22,101,46,96),(23,124,47,86),(24,114,48,76),(25,104,49,99),(26,127,50,89),(27,117,51,79),(28,107,52,69),(29,130,53,92),(30,120,54,82),(31,110,55,72),(32,100,56,95),(33,123,57,85)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33E | 44A | ··· | 44J | 66A | ··· | 66E |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 |
| size | 1 | 1 | 33 | 33 | 2 | 3 | 3 | 11 | 11 | 2 | 2 | ··· | 2 | 22 | 22 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D11 | C4×S3 | D22 | C4×D11 | S3×D11 | D33⋊C4 |
| kernel | D33⋊C4 | C11×Dic3 | C3×Dic11 | D66 | D33 | Dic11 | C22 | Dic3 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 5 | 2 | 5 | 10 | 5 | 5 |
Matrix representation of D33⋊C4 ►in GL4(𝔽397) generated by
| 140 | 185 | 0 | 0 |
| 15 | 224 | 0 | 0 |
| 0 | 0 | 395 | 76 |
| 0 | 0 | 141 | 1 |
| 334 | 289 | 0 | 0 |
| 272 | 63 | 0 | 0 |
| 0 | 0 | 396 | 0 |
| 0 | 0 | 141 | 1 |
| 396 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 334 | 24 |
| 0 | 0 | 0 | 63 |
G:=sub<GL(4,GF(397))| [140,15,0,0,185,224,0,0,0,0,395,141,0,0,76,1],[334,272,0,0,289,63,0,0,0,0,396,141,0,0,0,1],[396,0,0,0,0,396,0,0,0,0,334,0,0,0,24,63] >;
D33⋊C4 in GAP, Magma, Sage, TeX
D_{33}\rtimes C_4 % in TeX
G:=Group("D33:C4"); // GroupNames label
G:=SmallGroup(264,7);
// by ID
G=gap.SmallGroup(264,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,20,26,168,6004]);
// Polycyclic
G:=Group<a,b,c|a^33=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^23,c*b*c^-1=a^22*b>;
// generators/relations
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