direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C11×C3⋊D4, C33⋊9D4, D6⋊2C22, Dic3⋊C22, C22.17D6, C66.22C22, C3⋊2(D4×C11), (C2×C22)⋊3S3, (C2×C66)⋊6C2, (C2×C6)⋊2C22, (S3×C22)⋊5C2, C2.5(S3×C22), C6.5(C2×C22), C22⋊2(S3×C11), (C11×Dic3)⋊4C2, SmallGroup(264,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C3⋊D4
G = < a,b,c,d | a11=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C11×C3⋊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 80 90)(2 81 91)(3 82 92)(4 83 93)(5 84 94)(6 85 95)(7 86 96)(8 87 97)(9 88 98)(10 78 99)(11 79 89)(12 40 50)(13 41 51)(14 42 52)(15 43 53)(16 44 54)(17 34 55)(18 35 45)(19 36 46)(20 37 47)(21 38 48)(22 39 49)(23 122 60)(24 123 61)(25 124 62)(26 125 63)(27 126 64)(28 127 65)(29 128 66)(30 129 56)(31 130 57)(32 131 58)(33 132 59)(67 121 110)(68 111 100)(69 112 101)(70 113 102)(71 114 103)(72 115 104)(73 116 105)(74 117 106)(75 118 107)(76 119 108)(77 120 109)
(1 41 75 23)(2 42 76 24)(3 43 77 25)(4 44 67 26)(5 34 68 27)(6 35 69 28)(7 36 70 29)(8 37 71 30)(9 38 72 31)(10 39 73 32)(11 40 74 33)(12 117 59 79)(13 118 60 80)(14 119 61 81)(15 120 62 82)(16 121 63 83)(17 111 64 84)(18 112 65 85)(19 113 66 86)(20 114 56 87)(21 115 57 88)(22 116 58 78)(45 101 127 95)(46 102 128 96)(47 103 129 97)(48 104 130 98)(49 105 131 99)(50 106 132 89)(51 107 122 90)(52 108 123 91)(53 109 124 92)(54 110 125 93)(55 100 126 94)
(12 132)(13 122)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 131)(23 41)(24 42)(25 43)(26 44)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(33 40)(45 65)(46 66)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(78 99)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(100 111)(101 112)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)(109 120)(110 121)
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), (1,80,90)(2,81,91)(3,82,92)(4,83,93)(5,84,94)(6,85,95)(7,86,96)(8,87,97)(9,88,98)(10,78,99)(11,79,89)(12,40,50)(13,41,51)(14,42,52)(15,43,53)(16,44,54)(17,34,55)(18,35,45)(19,36,46)(20,37,47)(21,38,48)(22,39,49)(23,122,60)(24,123,61)(25,124,62)(26,125,63)(27,126,64)(28,127,65)(29,128,66)(30,129,56)(31,130,57)(32,131,58)(33,132,59)(67,121,110)(68,111,100)(69,112,101)(70,113,102)(71,114,103)(72,115,104)(73,116,105)(74,117,106)(75,118,107)(76,119,108)(77,120,109), (1,41,75,23)(2,42,76,24)(3,43,77,25)(4,44,67,26)(5,34,68,27)(6,35,69,28)(7,36,70,29)(8,37,71,30)(9,38,72,31)(10,39,73,32)(11,40,74,33)(12,117,59,79)(13,118,60,80)(14,119,61,81)(15,120,62,82)(16,121,63,83)(17,111,64,84)(18,112,65,85)(19,113,66,86)(20,114,56,87)(21,115,57,88)(22,116,58,78)(45,101,127,95)(46,102,128,96)(47,103,129,97)(48,104,130,98)(49,105,131,99)(50,106,132,89)(51,107,122,90)(52,108,123,91)(53,109,124,92)(54,110,125,93)(55,100,126,94), (12,132)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,41)(24,42)(25,43)(26,44)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(45,65)(46,66)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(78,99)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(100,111)(101,112)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)>;
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), (1,80,90)(2,81,91)(3,82,92)(4,83,93)(5,84,94)(6,85,95)(7,86,96)(8,87,97)(9,88,98)(10,78,99)(11,79,89)(12,40,50)(13,41,51)(14,42,52)(15,43,53)(16,44,54)(17,34,55)(18,35,45)(19,36,46)(20,37,47)(21,38,48)(22,39,49)(23,122,60)(24,123,61)(25,124,62)(26,125,63)(27,126,64)(28,127,65)(29,128,66)(30,129,56)(31,130,57)(32,131,58)(33,132,59)(67,121,110)(68,111,100)(69,112,101)(70,113,102)(71,114,103)(72,115,104)(73,116,105)(74,117,106)(75,118,107)(76,119,108)(77,120,109), (1,41,75,23)(2,42,76,24)(3,43,77,25)(4,44,67,26)(5,34,68,27)(6,35,69,28)(7,36,70,29)(8,37,71,30)(9,38,72,31)(10,39,73,32)(11,40,74,33)(12,117,59,79)(13,118,60,80)(14,119,61,81)(15,120,62,82)(16,121,63,83)(17,111,64,84)(18,112,65,85)(19,113,66,86)(20,114,56,87)(21,115,57,88)(22,116,58,78)(45,101,127,95)(46,102,128,96)(47,103,129,97)(48,104,130,98)(49,105,131,99)(50,106,132,89)(51,107,122,90)(52,108,123,91)(53,109,124,92)(54,110,125,93)(55,100,126,94), (12,132)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,41)(24,42)(25,43)(26,44)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(45,65)(46,66)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(78,99)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(100,111)(101,112)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121) );
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)], [(1,80,90),(2,81,91),(3,82,92),(4,83,93),(5,84,94),(6,85,95),(7,86,96),(8,87,97),(9,88,98),(10,78,99),(11,79,89),(12,40,50),(13,41,51),(14,42,52),(15,43,53),(16,44,54),(17,34,55),(18,35,45),(19,36,46),(20,37,47),(21,38,48),(22,39,49),(23,122,60),(24,123,61),(25,124,62),(26,125,63),(27,126,64),(28,127,65),(29,128,66),(30,129,56),(31,130,57),(32,131,58),(33,132,59),(67,121,110),(68,111,100),(69,112,101),(70,113,102),(71,114,103),(72,115,104),(73,116,105),(74,117,106),(75,118,107),(76,119,108),(77,120,109)], [(1,41,75,23),(2,42,76,24),(3,43,77,25),(4,44,67,26),(5,34,68,27),(6,35,69,28),(7,36,70,29),(8,37,71,30),(9,38,72,31),(10,39,73,32),(11,40,74,33),(12,117,59,79),(13,118,60,80),(14,119,61,81),(15,120,62,82),(16,121,63,83),(17,111,64,84),(18,112,65,85),(19,113,66,86),(20,114,56,87),(21,115,57,88),(22,116,58,78),(45,101,127,95),(46,102,128,96),(47,103,129,97),(48,104,130,98),(49,105,131,99),(50,106,132,89),(51,107,122,90),(52,108,123,91),(53,109,124,92),(54,110,125,93),(55,100,126,94)], [(12,132),(13,122),(14,123),(15,124),(16,125),(17,126),(18,127),(19,128),(20,129),(21,130),(22,131),(23,41),(24,42),(25,43),(26,44),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(33,40),(45,65),(46,66),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(78,99),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(100,111),(101,112),(102,113),(103,114),(104,115),(105,116),(106,117),(107,118),(108,119),(109,120),(110,121)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 22U | ··· | 22AD | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 |
| size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | S3 | D4 | D6 | C3⋊D4 | S3×C11 | D4×C11 | S3×C22 | C11×C3⋊D4 |
| kernel | C11×C3⋊D4 | C11×Dic3 | S3×C22 | C2×C66 | C3⋊D4 | Dic3 | D6 | C2×C6 | C2×C22 | C33 | C22 | C11 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 2 | 10 | 10 | 10 | 20 |
Matrix representation of C11×C3⋊D4 ►in GL2(𝔽397) generated by
| 273 | 0 |
| 0 | 273 |
| 0 | 396 |
| 1 | 396 |
| 351 | 23 |
| 374 | 46 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(397))| [273,0,0,273],[0,1,396,396],[351,374,23,46],[0,1,1,0] >;
C11×C3⋊D4 in GAP, Magma, Sage, TeX
C_{11}\times C_3\rtimes D_4 % in TeX
G:=Group("C11xC3:D4"); // GroupNames label
G:=SmallGroup(264,22);
// by ID
G=gap.SmallGroup(264,22);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-3,461,4404]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export