metabelian, supersoluble, monomial
Aliases: C122⋊16C2, C62.216C23, (C4×C12)⋊11S3, C12.81(C4×S3), C42⋊2(C3⋊S3), (C2×C12).424D6, C3⋊4(C42⋊2S3), C6.93(C4○D12), (C6×C12).285C22, C6.Dic6⋊28C2, C32⋊12(C42⋊C2), C6.11D12.12C2, C2.2(C12.59D6), (C4×C3⋊S3)⋊7C4, C6.63(S3×C2×C4), C4.22(C4×C3⋊S3), (C4×C3⋊Dic3)⋊19C2, (C3×C12).115(C2×C4), C3⋊Dic3.46(C2×C4), (C3×C6).94(C22×C4), (C3×C6).109(C4○D4), (C2×C6).233(C22×S3), C22.10(C22×C3⋊S3), (C22×C3⋊S3).79C22, (C2×C3⋊Dic3).151C22, C2.5(C2×C4×C3⋊S3), (C2×C4×C3⋊S3).21C2, (C2×C4).64(C2×C3⋊S3), (C2×C3⋊S3).40(C2×C4), SmallGroup(288,729)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C122⋊16C2
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5, cbc=a6b5 >
Subgroups: 772 in 228 conjugacy classes, 85 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C42⋊C2, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C22×C3⋊S3, C42⋊2S3, C4×C3⋊Dic3, C6.Dic6, C6.11D12, C122, C2×C4×C3⋊S3, C122⋊16C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C3⋊S3, C4×S3, C22×S3, C42⋊C2, C2×C3⋊S3, S3×C2×C4, C4○D12, C4×C3⋊S3, C22×C3⋊S3, C42⋊2S3, C2×C4×C3⋊S3, C12.59D6, C122⋊16C2
►On 144 points
Generators in S144
(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)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 41 87 109 100 24 67 73 136 122 26 57)(2 42 88 110 101 13 68 74 137 123 27 58)(3 43 89 111 102 14 69 75 138 124 28 59)(4 44 90 112 103 15 70 76 139 125 29 60)(5 45 91 113 104 16 71 77 140 126 30 49)(6 46 92 114 105 17 72 78 141 127 31 50)(7 47 93 115 106 18 61 79 142 128 32 51)(8 48 94 116 107 19 62 80 143 129 33 52)(9 37 95 117 108 20 63 81 144 130 34 53)(10 38 96 118 97 21 64 82 133 131 35 54)(11 39 85 119 98 22 65 83 134 132 36 55)(12 40 86 120 99 23 66 84 135 121 25 56)
(2 6)(3 11)(5 9)(8 12)(13 40)(14 45)(15 38)(16 43)(17 48)(18 41)(19 46)(20 39)(21 44)(22 37)(23 42)(24 47)(25 94)(26 87)(27 92)(28 85)(29 90)(30 95)(31 88)(32 93)(33 86)(34 91)(35 96)(36 89)(49 75)(50 80)(51 73)(52 78)(53 83)(54 76)(55 81)(56 74)(57 79)(58 84)(59 77)(60 82)(62 66)(63 71)(65 69)(68 72)(97 133)(98 138)(99 143)(100 136)(101 141)(102 134)(103 139)(104 144)(105 137)(106 142)(107 135)(108 140)(109 115)(110 120)(111 113)(112 118)(114 116)(117 119)(121 123)(122 128)(124 126)(125 131)(127 129)(130 132)
G:=sub<Sym(144)| (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)(133,134,135,136,137,138,139,140,141,142,143,144), (1,41,87,109,100,24,67,73,136,122,26,57)(2,42,88,110,101,13,68,74,137,123,27,58)(3,43,89,111,102,14,69,75,138,124,28,59)(4,44,90,112,103,15,70,76,139,125,29,60)(5,45,91,113,104,16,71,77,140,126,30,49)(6,46,92,114,105,17,72,78,141,127,31,50)(7,47,93,115,106,18,61,79,142,128,32,51)(8,48,94,116,107,19,62,80,143,129,33,52)(9,37,95,117,108,20,63,81,144,130,34,53)(10,38,96,118,97,21,64,82,133,131,35,54)(11,39,85,119,98,22,65,83,134,132,36,55)(12,40,86,120,99,23,66,84,135,121,25,56), (2,6)(3,11)(5,9)(8,12)(13,40)(14,45)(15,38)(16,43)(17,48)(18,41)(19,46)(20,39)(21,44)(22,37)(23,42)(24,47)(25,94)(26,87)(27,92)(28,85)(29,90)(30,95)(31,88)(32,93)(33,86)(34,91)(35,96)(36,89)(49,75)(50,80)(51,73)(52,78)(53,83)(54,76)(55,81)(56,74)(57,79)(58,84)(59,77)(60,82)(62,66)(63,71)(65,69)(68,72)(97,133)(98,138)(99,143)(100,136)(101,141)(102,134)(103,139)(104,144)(105,137)(106,142)(107,135)(108,140)(109,115)(110,120)(111,113)(112,118)(114,116)(117,119)(121,123)(122,128)(124,126)(125,131)(127,129)(130,132)>;
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)(133,134,135,136,137,138,139,140,141,142,143,144), (1,41,87,109,100,24,67,73,136,122,26,57)(2,42,88,110,101,13,68,74,137,123,27,58)(3,43,89,111,102,14,69,75,138,124,28,59)(4,44,90,112,103,15,70,76,139,125,29,60)(5,45,91,113,104,16,71,77,140,126,30,49)(6,46,92,114,105,17,72,78,141,127,31,50)(7,47,93,115,106,18,61,79,142,128,32,51)(8,48,94,116,107,19,62,80,143,129,33,52)(9,37,95,117,108,20,63,81,144,130,34,53)(10,38,96,118,97,21,64,82,133,131,35,54)(11,39,85,119,98,22,65,83,134,132,36,55)(12,40,86,120,99,23,66,84,135,121,25,56), (2,6)(3,11)(5,9)(8,12)(13,40)(14,45)(15,38)(16,43)(17,48)(18,41)(19,46)(20,39)(21,44)(22,37)(23,42)(24,47)(25,94)(26,87)(27,92)(28,85)(29,90)(30,95)(31,88)(32,93)(33,86)(34,91)(35,96)(36,89)(49,75)(50,80)(51,73)(52,78)(53,83)(54,76)(55,81)(56,74)(57,79)(58,84)(59,77)(60,82)(62,66)(63,71)(65,69)(68,72)(97,133)(98,138)(99,143)(100,136)(101,141)(102,134)(103,139)(104,144)(105,137)(106,142)(107,135)(108,140)(109,115)(110,120)(111,113)(112,118)(114,116)(117,119)(121,123)(122,128)(124,126)(125,131)(127,129)(130,132) );
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),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,41,87,109,100,24,67,73,136,122,26,57),(2,42,88,110,101,13,68,74,137,123,27,58),(3,43,89,111,102,14,69,75,138,124,28,59),(4,44,90,112,103,15,70,76,139,125,29,60),(5,45,91,113,104,16,71,77,140,126,30,49),(6,46,92,114,105,17,72,78,141,127,31,50),(7,47,93,115,106,18,61,79,142,128,32,51),(8,48,94,116,107,19,62,80,143,129,33,52),(9,37,95,117,108,20,63,81,144,130,34,53),(10,38,96,118,97,21,64,82,133,131,35,54),(11,39,85,119,98,22,65,83,134,132,36,55),(12,40,86,120,99,23,66,84,135,121,25,56)], [(2,6),(3,11),(5,9),(8,12),(13,40),(14,45),(15,38),(16,43),(17,48),(18,41),(19,46),(20,39),(21,44),(22,37),(23,42),(24,47),(25,94),(26,87),(27,92),(28,85),(29,90),(30,95),(31,88),(32,93),(33,86),(34,91),(35,96),(36,89),(49,75),(50,80),(51,73),(52,78),(53,83),(54,76),(55,81),(56,74),(57,79),(58,84),(59,77),(60,82),(62,66),(63,71),(65,69),(68,72),(97,133),(98,138),(99,143),(100,136),(101,141),(102,134),(103,139),(104,144),(105,137),(106,142),(107,135),(108,140),(109,115),(110,120),(111,113),(112,118),(114,116),(117,119),(121,123),(122,128),(124,126),(125,131),(127,129),(130,132)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | C4○D4 | C4×S3 | C4○D12 |
| kernel | C122⋊16C2 | C4×C3⋊Dic3 | C6.Dic6 | C6.11D12 | C122 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C4×C12 | C2×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 4 | 12 | 4 | 16 | 32 |
Matrix representation of C122⋊16C2 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 5 |
| 0 | 0 | 0 | 8 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 0 | 0 |
| 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 9 | 2 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8,8,0,0,0,5,0],[5,0,0,0,0,0,2,11,0,0,0,2,4,0,0,0,0,0,11,9,0,0,0,4,2],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
C122⋊16C2 in GAP, Magma, Sage, TeX
C_{12}^2\rtimes_{16}C_2 % in TeX
G:=Group("C12^2:16C2"); // GroupNames label
G:=SmallGroup(288,729);
// by ID
G=gap.SmallGroup(288,729);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=a^6*b^5>;
// generators/relations