metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C33⋊7D4, D66⋊2C2, C2.5D66, C22.12D6, C6.12D22, C22⋊2D33, Dic33⋊1C2, C66.12C22, (C2×C22)⋊4S3, (C2×C66)⋊2C2, (C2×C6)⋊2D11, C11⋊3(C3⋊D4), C3⋊3(C11⋊D4), SmallGroup(264,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊7D4
G = < a,b,c | a33=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C33⋊7D4
►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 83 46 128)(2 82 47 127)(3 81 48 126)(4 80 49 125)(5 79 50 124)(6 78 51 123)(7 77 52 122)(8 76 53 121)(9 75 54 120)(10 74 55 119)(11 73 56 118)(12 72 57 117)(13 71 58 116)(14 70 59 115)(15 69 60 114)(16 68 61 113)(17 67 62 112)(18 99 63 111)(19 98 64 110)(20 97 65 109)(21 96 66 108)(22 95 34 107)(23 94 35 106)(24 93 36 105)(25 92 37 104)(26 91 38 103)(27 90 39 102)(28 89 40 101)(29 88 41 100)(30 87 42 132)(31 86 43 131)(32 85 44 130)(33 84 45 129)
(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 18)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(59 66)(60 65)(61 64)(62 63)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)
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), (1,83,46,128)(2,82,47,127)(3,81,48,126)(4,80,49,125)(5,79,50,124)(6,78,51,123)(7,77,52,122)(8,76,53,121)(9,75,54,120)(10,74,55,119)(11,73,56,118)(12,72,57,117)(13,71,58,116)(14,70,59,115)(15,69,60,114)(16,68,61,113)(17,67,62,112)(18,99,63,111)(19,98,64,110)(20,97,65,109)(21,96,66,108)(22,95,34,107)(23,94,35,106)(24,93,36,105)(25,92,37,104)(26,91,38,103)(27,90,39,102)(28,89,40,101)(29,88,41,100)(30,87,42,132)(31,86,43,131)(32,85,44,130)(33,84,45,129), (2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(59,66)(60,65)(61,64)(62,63)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)>;
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), (1,83,46,128)(2,82,47,127)(3,81,48,126)(4,80,49,125)(5,79,50,124)(6,78,51,123)(7,77,52,122)(8,76,53,121)(9,75,54,120)(10,74,55,119)(11,73,56,118)(12,72,57,117)(13,71,58,116)(14,70,59,115)(15,69,60,114)(16,68,61,113)(17,67,62,112)(18,99,63,111)(19,98,64,110)(20,97,65,109)(21,96,66,108)(22,95,34,107)(23,94,35,106)(24,93,36,105)(25,92,37,104)(26,91,38,103)(27,90,39,102)(28,89,40,101)(29,88,41,100)(30,87,42,132)(31,86,43,131)(32,85,44,130)(33,84,45,129), (2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(59,66)(60,65)(61,64)(62,63)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112) );
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)], [(1,83,46,128),(2,82,47,127),(3,81,48,126),(4,80,49,125),(5,79,50,124),(6,78,51,123),(7,77,52,122),(8,76,53,121),(9,75,54,120),(10,74,55,119),(11,73,56,118),(12,72,57,117),(13,71,58,116),(14,70,59,115),(15,69,60,114),(16,68,61,113),(17,67,62,112),(18,99,63,111),(19,98,64,110),(20,97,65,109),(21,96,66,108),(22,95,34,107),(23,94,35,106),(24,93,36,105),(25,92,37,104),(26,91,38,103),(27,90,39,102),(28,89,40,101),(29,88,41,100),(30,87,42,132),(31,86,43,131),(32,85,44,130),(33,84,45,129)], [(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,18),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(59,66),(60,65),(61,64),(62,63),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,132),(80,131),(81,130),(82,129),(83,128),(84,127),(85,126),(86,125),(87,124),(88,123),(89,122),(90,121),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112)]])
69 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 11A | ··· | 11E | 22A | ··· | 22O | 33A | ··· | 33J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 2 | 66 | 2 | 66 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D11 | C3⋊D4 | D22 | D33 | C11⋊D4 | D66 | C33⋊7D4 |
| kernel | C33⋊7D4 | Dic33 | D66 | C2×C66 | C2×C22 | C33 | C22 | C2×C6 | C11 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of C33⋊7D4 ►in GL2(𝔽397) generated by
| 15 | 40 |
| 361 | 354 |
| 96 | 265 |
| 139 | 301 |
| 153 | 217 |
| 33 | 244 |
G:=sub<GL(2,GF(397))| [15,361,40,354],[96,139,265,301],[153,33,217,244] >;
C33⋊7D4 in GAP, Magma, Sage, TeX
C_{33}\rtimes_7D_4 % in TeX
G:=Group("C33:7D4"); // GroupNames label
G:=SmallGroup(264,27);
// by ID
G=gap.SmallGroup(264,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,61,323,6004]);
// Polycyclic
G:=Group<a,b,c|a^33=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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