direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C33, C4⋊C66, C44⋊3C6, C12⋊3C22, C132⋊7C2, C22⋊2C66, C66.23C22, (C2×C6)⋊1C22, (C2×C66)⋊1C2, (C2×C22)⋊3C6, C6.6(C2×C22), C22.6(C2×C6), C2.1(C2×C66), SmallGroup(264,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C33
G = < a,b,c | a33=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C33
►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)
(1 102 98 59)(2 103 99 60)(3 104 67 61)(4 105 68 62)(5 106 69 63)(6 107 70 64)(7 108 71 65)(8 109 72 66)(9 110 73 34)(10 111 74 35)(11 112 75 36)(12 113 76 37)(13 114 77 38)(14 115 78 39)(15 116 79 40)(16 117 80 41)(17 118 81 42)(18 119 82 43)(19 120 83 44)(20 121 84 45)(21 122 85 46)(22 123 86 47)(23 124 87 48)(24 125 88 49)(25 126 89 50)(26 127 90 51)(27 128 91 52)(28 129 92 53)(29 130 93 54)(30 131 94 55)(31 132 95 56)(32 100 96 57)(33 101 97 58)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 100)(97 101)(98 102)(99 103)
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), (1,102,98,59)(2,103,99,60)(3,104,67,61)(4,105,68,62)(5,106,69,63)(6,107,70,64)(7,108,71,65)(8,109,72,66)(9,110,73,34)(10,111,74,35)(11,112,75,36)(12,113,76,37)(13,114,77,38)(14,115,78,39)(15,116,79,40)(16,117,80,41)(17,118,81,42)(18,119,82,43)(19,120,83,44)(20,121,84,45)(21,122,85,46)(22,123,86,47)(23,124,87,48)(24,125,88,49)(25,126,89,50)(26,127,90,51)(27,128,91,52)(28,129,92,53)(29,130,93,54)(30,131,94,55)(31,132,95,56)(32,100,96,57)(33,101,97,58), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,100)(97,101)(98,102)(99,103)>;
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), (1,102,98,59)(2,103,99,60)(3,104,67,61)(4,105,68,62)(5,106,69,63)(6,107,70,64)(7,108,71,65)(8,109,72,66)(9,110,73,34)(10,111,74,35)(11,112,75,36)(12,113,76,37)(13,114,77,38)(14,115,78,39)(15,116,79,40)(16,117,80,41)(17,118,81,42)(18,119,82,43)(19,120,83,44)(20,121,84,45)(21,122,85,46)(22,123,86,47)(23,124,87,48)(24,125,88,49)(25,126,89,50)(26,127,90,51)(27,128,91,52)(28,129,92,53)(29,130,93,54)(30,131,94,55)(31,132,95,56)(32,100,96,57)(33,101,97,58), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,100)(97,101)(98,102)(99,103) );
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)], [(1,102,98,59),(2,103,99,60),(3,104,67,61),(4,105,68,62),(5,106,69,63),(6,107,70,64),(7,108,71,65),(8,109,72,66),(9,110,73,34),(10,111,74,35),(11,112,75,36),(12,113,76,37),(13,114,77,38),(14,115,78,39),(15,116,79,40),(16,117,80,41),(17,118,81,42),(18,119,82,43),(19,120,83,44),(20,121,84,45),(21,122,85,46),(22,123,86,47),(23,124,87,48),(24,125,88,49),(25,126,89,50),(26,127,90,51),(27,128,91,52),(28,129,92,53),(29,130,93,54),(30,131,94,55),(31,132,95,56),(32,100,96,57),(33,101,97,58)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,121),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,129),(93,130),(94,131),(95,132),(96,100),(97,101),(98,102),(99,103)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 11A | ··· | 11J | 12A | 12B | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33T | 44A | ··· | 44J | 66A | ··· | 66T | 66U | ··· | 66BH | 132A | ··· | 132T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C11 | C22 | C22 | C33 | C66 | C66 | D4 | C3×D4 | D4×C11 | D4×C33 |
| kernel | D4×C33 | C132 | C2×C66 | D4×C11 | C44 | C2×C22 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C33 | C11 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 10 | 10 | 20 | 20 | 20 | 40 | 1 | 2 | 10 | 20 |
Matrix representation of D4×C33 ►in GL2(𝔽397) generated by
| 147 | 0 |
| 0 | 147 |
| 0 | 1 |
| 396 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(397))| [147,0,0,147],[0,396,1,0],[0,1,1,0] >;
D4×C33 in GAP, Magma, Sage, TeX
D_4\times C_{33} % in TeX
G:=Group("D4xC33"); // GroupNames label
G:=SmallGroup(264,29);
// by ID
G=gap.SmallGroup(264,29);
# by ID
G:=PCGroup([5,-2,-2,-3,-11,-2,1341]);
// Polycyclic
G:=Group<a,b,c|a^33=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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