metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C33⋊2D4, C3⋊2D44, D22⋊2S3, D66⋊3C2, Dic3⋊D11, C22.5D6, C6.5D22, C66.5C22, (C6×D11)⋊2C2, C11⋊1(C3⋊D4), C2.5(S3×D11), (C11×Dic3)⋊3C2, SmallGroup(264,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D44
G = < a,b,c | a3=b44=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D44
►On 132 points
Generators in S132
(1 62 93)(2 94 63)(3 64 95)(4 96 65)(5 66 97)(6 98 67)(7 68 99)(8 100 69)(9 70 101)(10 102 71)(11 72 103)(12 104 73)(13 74 105)(14 106 75)(15 76 107)(16 108 77)(17 78 109)(18 110 79)(19 80 111)(20 112 81)(21 82 113)(22 114 83)(23 84 115)(24 116 85)(25 86 117)(26 118 87)(27 88 119)(28 120 45)(29 46 121)(30 122 47)(31 48 123)(32 124 49)(33 50 125)(34 126 51)(35 52 127)(36 128 53)(37 54 129)(38 130 55)(39 56 131)(40 132 57)(41 58 89)(42 90 59)(43 60 91)(44 92 61)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 132)(78 131)(79 130)(80 129)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)
G:=sub<Sym(132)| (1,62,93)(2,94,63)(3,64,95)(4,96,65)(5,66,97)(6,98,67)(7,68,99)(8,100,69)(9,70,101)(10,102,71)(11,72,103)(12,104,73)(13,74,105)(14,106,75)(15,76,107)(16,108,77)(17,78,109)(18,110,79)(19,80,111)(20,112,81)(21,82,113)(22,114,83)(23,84,115)(24,116,85)(25,86,117)(26,118,87)(27,88,119)(28,120,45)(29,46,121)(30,122,47)(31,48,123)(32,124,49)(33,50,125)(34,126,51)(35,52,127)(36,128,53)(37,54,129)(38,130,55)(39,56,131)(40,132,57)(41,58,89)(42,90,59)(43,60,91)(44,92,61), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)>;
G:=Group( (1,62,93)(2,94,63)(3,64,95)(4,96,65)(5,66,97)(6,98,67)(7,68,99)(8,100,69)(9,70,101)(10,102,71)(11,72,103)(12,104,73)(13,74,105)(14,106,75)(15,76,107)(16,108,77)(17,78,109)(18,110,79)(19,80,111)(20,112,81)(21,82,113)(22,114,83)(23,84,115)(24,116,85)(25,86,117)(26,118,87)(27,88,119)(28,120,45)(29,46,121)(30,122,47)(31,48,123)(32,124,49)(33,50,125)(34,126,51)(35,52,127)(36,128,53)(37,54,129)(38,130,55)(39,56,131)(40,132,57)(41,58,89)(42,90,59)(43,60,91)(44,92,61), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121) );
G=PermutationGroup([[(1,62,93),(2,94,63),(3,64,95),(4,96,65),(5,66,97),(6,98,67),(7,68,99),(8,100,69),(9,70,101),(10,102,71),(11,72,103),(12,104,73),(13,74,105),(14,106,75),(15,76,107),(16,108,77),(17,78,109),(18,110,79),(19,80,111),(20,112,81),(21,82,113),(22,114,83),(23,84,115),(24,116,85),(25,86,117),(26,118,87),(27,88,119),(28,120,45),(29,46,121),(30,122,47),(31,48,123),(32,124,49),(33,50,125),(34,126,51),(35,52,127),(36,128,53),(37,54,129),(38,130,55),(39,56,131),(40,132,57),(41,58,89),(42,90,59),(43,60,91),(44,92,61)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(45,120),(46,119),(47,118),(48,117),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,108),(58,107),(59,106),(60,105),(61,104),(62,103),(63,102),(64,101),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,132),(78,131),(79,130),(80,129),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 11A | ··· | 11E | 22A | ··· | 22E | 33A | ··· | 33E | 44A | ··· | 44J | 66A | ··· | 66E |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 |
| size | 1 | 1 | 22 | 66 | 2 | 6 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 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 | D44 | S3×D11 | C3⋊D44 |
| kernel | C3⋊D44 | C11×Dic3 | C6×D11 | D66 | D22 | C33 | C22 | Dic3 | C11 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 5 | 5 |
Matrix representation of C3⋊D44 ►in GL4(𝔽397) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 173 |
| 0 | 0 | 257 | 395 |
| 185 | 247 | 0 | 0 |
| 150 | 108 | 0 | 0 |
| 0 | 0 | 69 | 18 |
| 0 | 0 | 353 | 328 |
| 185 | 247 | 0 | 0 |
| 117 | 212 | 0 | 0 |
| 0 | 0 | 396 | 0 |
| 0 | 0 | 140 | 1 |
G:=sub<GL(4,GF(397))| [1,0,0,0,0,1,0,0,0,0,1,257,0,0,173,395],[185,150,0,0,247,108,0,0,0,0,69,353,0,0,18,328],[185,117,0,0,247,212,0,0,0,0,396,140,0,0,0,1] >;
C3⋊D44 in GAP, Magma, Sage, TeX
C_3\rtimes D_{44} % in TeX
G:=Group("C3:D44"); // GroupNames label
G:=SmallGroup(264,9);
// by ID
G=gap.SmallGroup(264,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,61,26,168,6004]);
// Polycyclic
G:=Group<a,b,c|a^3=b^44=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export