metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C33⋊1D4, D6⋊1D11, D22⋊1S3, C6.4D22, C22.4D6, Dic33⋊4C2, C66.4C22, (S3×C22)⋊1C2, (C6×D11)⋊1C2, C11⋊2(C3⋊D4), C3⋊2(C11⋊D4), C2.4(S3×D11), SmallGroup(264,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊D4
G = < a,b,c | a33=b4=c2=1, bab-1=a-1, cac=a10, cbc=b-1 >
Smallest permutation representation of C33⋊D4
►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 82 54 101)(2 81 55 100)(3 80 56 132)(4 79 57 131)(5 78 58 130)(6 77 59 129)(7 76 60 128)(8 75 61 127)(9 74 62 126)(10 73 63 125)(11 72 64 124)(12 71 65 123)(13 70 66 122)(14 69 34 121)(15 68 35 120)(16 67 36 119)(17 99 37 118)(18 98 38 117)(19 97 39 116)(20 96 40 115)(21 95 41 114)(22 94 42 113)(23 93 43 112)(24 92 44 111)(25 91 45 110)(26 90 46 109)(27 89 47 108)(28 88 48 107)(29 87 49 106)(30 86 50 105)(31 85 51 104)(32 84 52 103)(33 83 53 102)
(2 11)(3 21)(4 31)(5 8)(6 18)(7 28)(9 15)(10 25)(13 22)(14 32)(16 19)(17 29)(20 26)(24 33)(27 30)(34 52)(35 62)(36 39)(37 49)(38 59)(40 46)(41 56)(42 66)(44 53)(45 63)(47 50)(48 60)(51 57)(55 64)(58 61)(67 116)(68 126)(69 103)(70 113)(71 123)(72 100)(73 110)(74 120)(75 130)(76 107)(77 117)(78 127)(79 104)(80 114)(81 124)(82 101)(83 111)(84 121)(85 131)(86 108)(87 118)(88 128)(89 105)(90 115)(91 125)(92 102)(93 112)(94 122)(95 132)(96 109)(97 119)(98 129)(99 106)
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), (1,82,54,101)(2,81,55,100)(3,80,56,132)(4,79,57,131)(5,78,58,130)(6,77,59,129)(7,76,60,128)(8,75,61,127)(9,74,62,126)(10,73,63,125)(11,72,64,124)(12,71,65,123)(13,70,66,122)(14,69,34,121)(15,68,35,120)(16,67,36,119)(17,99,37,118)(18,98,38,117)(19,97,39,116)(20,96,40,115)(21,95,41,114)(22,94,42,113)(23,93,43,112)(24,92,44,111)(25,91,45,110)(26,90,46,109)(27,89,47,108)(28,88,48,107)(29,87,49,106)(30,86,50,105)(31,85,51,104)(32,84,52,103)(33,83,53,102), (2,11)(3,21)(4,31)(5,8)(6,18)(7,28)(9,15)(10,25)(13,22)(14,32)(16,19)(17,29)(20,26)(24,33)(27,30)(34,52)(35,62)(36,39)(37,49)(38,59)(40,46)(41,56)(42,66)(44,53)(45,63)(47,50)(48,60)(51,57)(55,64)(58,61)(67,116)(68,126)(69,103)(70,113)(71,123)(72,100)(73,110)(74,120)(75,130)(76,107)(77,117)(78,127)(79,104)(80,114)(81,124)(82,101)(83,111)(84,121)(85,131)(86,108)(87,118)(88,128)(89,105)(90,115)(91,125)(92,102)(93,112)(94,122)(95,132)(96,109)(97,119)(98,129)(99,106)>;
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), (1,82,54,101)(2,81,55,100)(3,80,56,132)(4,79,57,131)(5,78,58,130)(6,77,59,129)(7,76,60,128)(8,75,61,127)(9,74,62,126)(10,73,63,125)(11,72,64,124)(12,71,65,123)(13,70,66,122)(14,69,34,121)(15,68,35,120)(16,67,36,119)(17,99,37,118)(18,98,38,117)(19,97,39,116)(20,96,40,115)(21,95,41,114)(22,94,42,113)(23,93,43,112)(24,92,44,111)(25,91,45,110)(26,90,46,109)(27,89,47,108)(28,88,48,107)(29,87,49,106)(30,86,50,105)(31,85,51,104)(32,84,52,103)(33,83,53,102), (2,11)(3,21)(4,31)(5,8)(6,18)(7,28)(9,15)(10,25)(13,22)(14,32)(16,19)(17,29)(20,26)(24,33)(27,30)(34,52)(35,62)(36,39)(37,49)(38,59)(40,46)(41,56)(42,66)(44,53)(45,63)(47,50)(48,60)(51,57)(55,64)(58,61)(67,116)(68,126)(69,103)(70,113)(71,123)(72,100)(73,110)(74,120)(75,130)(76,107)(77,117)(78,127)(79,104)(80,114)(81,124)(82,101)(83,111)(84,121)(85,131)(86,108)(87,118)(88,128)(89,105)(90,115)(91,125)(92,102)(93,112)(94,122)(95,132)(96,109)(97,119)(98,129)(99,106) );
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)], [(1,82,54,101),(2,81,55,100),(3,80,56,132),(4,79,57,131),(5,78,58,130),(6,77,59,129),(7,76,60,128),(8,75,61,127),(9,74,62,126),(10,73,63,125),(11,72,64,124),(12,71,65,123),(13,70,66,122),(14,69,34,121),(15,68,35,120),(16,67,36,119),(17,99,37,118),(18,98,38,117),(19,97,39,116),(20,96,40,115),(21,95,41,114),(22,94,42,113),(23,93,43,112),(24,92,44,111),(25,91,45,110),(26,90,46,109),(27,89,47,108),(28,88,48,107),(29,87,49,106),(30,86,50,105),(31,85,51,104),(32,84,52,103),(33,83,53,102)], [(2,11),(3,21),(4,31),(5,8),(6,18),(7,28),(9,15),(10,25),(13,22),(14,32),(16,19),(17,29),(20,26),(24,33),(27,30),(34,52),(35,62),(36,39),(37,49),(38,59),(40,46),(41,56),(42,66),(44,53),(45,63),(47,50),(48,60),(51,57),(55,64),(58,61),(67,116),(68,126),(69,103),(70,113),(71,123),(72,100),(73,110),(74,120),(75,130),(76,107),(77,117),(78,127),(79,104),(80,114),(81,124),(82,101),(83,111),(84,121),(85,131),(86,108),(87,118),(88,128),(89,105),(90,115),(91,125),(92,102),(93,112),(94,122),(95,132),(96,109),(97,119),(98,129),(99,106)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22O | 33A | ··· | 33E | 66A | ··· | 66E |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 6 | 22 | 2 | 66 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D11 | C3⋊D4 | D22 | C11⋊D4 | S3×D11 | C33⋊D4 |
| kernel | C33⋊D4 | Dic33 | C6×D11 | S3×C22 | D22 | C33 | C22 | D6 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 5 | 5 |
Matrix representation of C33⋊D4 ►in GL4(𝔽397) generated by
| 362 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 301 | 45 |
| 0 | 0 | 352 | 385 |
| 0 | 396 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 293 | 396 |
| 1 | 0 | 0 | 0 |
| 0 | 396 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 293 | 396 |
G:=sub<GL(4,GF(397))| [362,0,0,0,0,34,0,0,0,0,301,352,0,0,45,385],[0,1,0,0,396,0,0,0,0,0,1,293,0,0,0,396],[1,0,0,0,0,396,0,0,0,0,1,293,0,0,0,396] >;
C33⋊D4 in GAP, Magma, Sage, TeX
C_{33}\rtimes D_4 % in TeX
G:=Group("C33:D4"); // GroupNames label
G:=SmallGroup(264,8);
// by ID
G=gap.SmallGroup(264,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,61,168,6004]);
// Polycyclic
G:=Group<a,b,c|a^33=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^10,c*b*c=b^-1>;
// generators/relations
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