direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D33, C44⋊2S3, C132⋊2C2, C12⋊2D11, C6.9D22, C2.1D66, C22.9D6, D66.2C2, Dic33⋊5C2, C66.9C22, C11⋊2(C4×S3), C33⋊4(C2×C4), C3⋊2(C4×D11), SmallGroup(264,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C4×D33 |
Generators and relations for C4×D33
G = < a,b,c | a4=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D33
►On 132 points
Generators in S132
(1 104 62 83)(2 105 63 84)(3 106 64 85)(4 107 65 86)(5 108 66 87)(6 109 34 88)(7 110 35 89)(8 111 36 90)(9 112 37 91)(10 113 38 92)(11 114 39 93)(12 115 40 94)(13 116 41 95)(14 117 42 96)(15 118 43 97)(16 119 44 98)(17 120 45 99)(18 121 46 67)(19 122 47 68)(20 123 48 69)(21 124 49 70)(22 125 50 71)(23 126 51 72)(24 127 52 73)(25 128 53 74)(26 129 54 75)(27 130 55 76)(28 131 56 77)(29 132 57 78)(30 100 58 79)(31 101 59 80)(32 102 60 81)(33 103 61 82)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 66)(30 65)(31 64)(32 63)(33 62)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 132)(88 131)(89 130)(90 129)(91 128)(92 127)(93 126)(94 125)(95 124)(96 123)(97 122)(98 121)(99 120)
G:=sub<Sym(132)| (1,104,62,83)(2,105,63,84)(3,106,64,85)(4,107,65,86)(5,108,66,87)(6,109,34,88)(7,110,35,89)(8,111,36,90)(9,112,37,91)(10,113,38,92)(11,114,39,93)(12,115,40,94)(13,116,41,95)(14,117,42,96)(15,118,43,97)(16,119,44,98)(17,120,45,99)(18,121,46,67)(19,122,47,68)(20,123,48,69)(21,124,49,70)(22,125,50,71)(23,126,51,72)(24,127,52,73)(25,128,53,74)(26,129,54,75)(27,130,55,76)(28,131,56,77)(29,132,57,78)(30,100,58,79)(31,101,59,80)(32,102,60,81)(33,103,61,82), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,66)(30,65)(31,64)(32,63)(33,62)(67,119)(68,118)(69,117)(70,116)(71,115)(72,114)(73,113)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,120)>;
G:=Group( (1,104,62,83)(2,105,63,84)(3,106,64,85)(4,107,65,86)(5,108,66,87)(6,109,34,88)(7,110,35,89)(8,111,36,90)(9,112,37,91)(10,113,38,92)(11,114,39,93)(12,115,40,94)(13,116,41,95)(14,117,42,96)(15,118,43,97)(16,119,44,98)(17,120,45,99)(18,121,46,67)(19,122,47,68)(20,123,48,69)(21,124,49,70)(22,125,50,71)(23,126,51,72)(24,127,52,73)(25,128,53,74)(26,129,54,75)(27,130,55,76)(28,131,56,77)(29,132,57,78)(30,100,58,79)(31,101,59,80)(32,102,60,81)(33,103,61,82), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,56)(7,55)(8,54)(9,53)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,66)(30,65)(31,64)(32,63)(33,62)(67,119)(68,118)(69,117)(70,116)(71,115)(72,114)(73,113)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,101)(86,100)(87,132)(88,131)(89,130)(90,129)(91,128)(92,127)(93,126)(94,125)(95,124)(96,123)(97,122)(98,121)(99,120) );
G=PermutationGroup([[(1,104,62,83),(2,105,63,84),(3,106,64,85),(4,107,65,86),(5,108,66,87),(6,109,34,88),(7,110,35,89),(8,111,36,90),(9,112,37,91),(10,113,38,92),(11,114,39,93),(12,115,40,94),(13,116,41,95),(14,117,42,96),(15,118,43,97),(16,119,44,98),(17,120,45,99),(18,121,46,67),(19,122,47,68),(20,123,48,69),(21,124,49,70),(22,125,50,71),(23,126,51,72),(24,127,52,73),(25,128,53,74),(26,129,54,75),(27,130,55,76),(28,131,56,77),(29,132,57,78),(30,100,58,79),(31,101,59,80),(32,102,60,81),(33,103,61,82)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,56),(7,55),(8,54),(9,53),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,66),(30,65),(31,64),(32,63),(33,62),(67,119),(68,118),(69,117),(70,116),(71,115),(72,114),(73,113),(74,112),(75,111),(76,110),(77,109),(78,108),(79,107),(80,106),(81,105),(82,104),(83,103),(84,102),(85,101),(86,100),(87,132),(88,131),(89,130),(90,129),(91,128),(92,127),(93,126),(94,125),(95,124),(96,123),(97,122),(98,121),(99,120)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 33 | 33 | 2 | 1 | 1 | 33 | 33 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D11 | C4×S3 | D22 | D33 | C4×D11 | D66 | C4×D33 |
| kernel | C4×D33 | Dic33 | C132 | D66 | D33 | C44 | C22 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C4×D33 ►in GL4(𝔽397) generated by
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 396 | 0 |
| 0 | 0 | 0 | 396 |
| 50 | 82 | 0 | 0 |
| 153 | 243 | 0 | 0 |
| 0 | 0 | 141 | 156 |
| 0 | 0 | 81 | 374 |
| 396 | 266 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 232 | 21 |
| 0 | 0 | 216 | 165 |
G:=sub<GL(4,GF(397))| [63,0,0,0,0,63,0,0,0,0,396,0,0,0,0,396],[50,153,0,0,82,243,0,0,0,0,141,81,0,0,156,374],[396,0,0,0,266,1,0,0,0,0,232,216,0,0,21,165] >;
C4×D33 in GAP, Magma, Sage, TeX
C_4\times D_{33} % in TeX
G:=Group("C4xD33"); // GroupNames label
G:=SmallGroup(264,24);
// by ID
G=gap.SmallGroup(264,24);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,26,323,6004]);
// Polycyclic
G:=Group<a,b,c|a^4=b^33=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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