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 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C32, Dic3 [×16], C12 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×16], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C42⋊C2, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3 [×4], Dic3⋊C4 [×8], D6⋊C4 [×8], C4×C12 [×4], S3×C2×C4 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3, C42⋊2S3 [×4], C4×C3⋊Dic3, C6.Dic6 [×2], C6.11D12 [×2], C122, C2×C4×C3⋊S3, C122⋊16C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C4○D4 [×2], C3⋊S3, C4×S3 [×8], C22×S3 [×4], C42⋊C2, C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4○D12 [×8], C4×C3⋊S3 [×2], C22×C3⋊S3, C42⋊2S3 [×4], C2×C4×C3⋊S3, C12.59D6 [×2], C122⋊16C2
►On 144 points
(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)
(1 136 17 76 126 55 45 65 94 102 25 114)(2 137 18 77 127 56 46 66 95 103 26 115)(3 138 19 78 128 57 47 67 96 104 27 116)(4 139 20 79 129 58 48 68 85 105 28 117)(5 140 21 80 130 59 37 69 86 106 29 118)(6 141 22 81 131 60 38 70 87 107 30 119)(7 142 23 82 132 49 39 71 88 108 31 120)(8 143 24 83 121 50 40 72 89 97 32 109)(9 144 13 84 122 51 41 61 90 98 33 110)(10 133 14 73 123 52 42 62 91 99 34 111)(11 134 15 74 124 53 43 63 92 100 35 112)(12 135 16 75 125 54 44 64 93 101 36 113)
(2 6)(3 11)(5 9)(8 12)(13 29)(14 34)(15 27)(16 32)(17 25)(18 30)(19 35)(20 28)(21 33)(22 26)(23 31)(24 36)(37 41)(38 46)(40 44)(43 47)(49 136)(50 141)(51 134)(52 139)(53 144)(54 137)(55 142)(56 135)(57 140)(58 133)(59 138)(60 143)(61 112)(62 117)(63 110)(64 115)(65 120)(66 113)(67 118)(68 111)(69 116)(70 109)(71 114)(72 119)(73 79)(74 84)(75 77)(76 82)(78 80)(81 83)(85 129)(86 122)(87 127)(88 132)(89 125)(90 130)(91 123)(92 128)(93 121)(94 126)(95 131)(96 124)(97 107)(98 100)(99 105)(101 103)(102 108)(104 106)
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), (1,136,17,76,126,55,45,65,94,102,25,114)(2,137,18,77,127,56,46,66,95,103,26,115)(3,138,19,78,128,57,47,67,96,104,27,116)(4,139,20,79,129,58,48,68,85,105,28,117)(5,140,21,80,130,59,37,69,86,106,29,118)(6,141,22,81,131,60,38,70,87,107,30,119)(7,142,23,82,132,49,39,71,88,108,31,120)(8,143,24,83,121,50,40,72,89,97,32,109)(9,144,13,84,122,51,41,61,90,98,33,110)(10,133,14,73,123,52,42,62,91,99,34,111)(11,134,15,74,124,53,43,63,92,100,35,112)(12,135,16,75,125,54,44,64,93,101,36,113), (2,6)(3,11)(5,9)(8,12)(13,29)(14,34)(15,27)(16,32)(17,25)(18,30)(19,35)(20,28)(21,33)(22,26)(23,31)(24,36)(37,41)(38,46)(40,44)(43,47)(49,136)(50,141)(51,134)(52,139)(53,144)(54,137)(55,142)(56,135)(57,140)(58,133)(59,138)(60,143)(61,112)(62,117)(63,110)(64,115)(65,120)(66,113)(67,118)(68,111)(69,116)(70,109)(71,114)(72,119)(73,79)(74,84)(75,77)(76,82)(78,80)(81,83)(85,129)(86,122)(87,127)(88,132)(89,125)(90,130)(91,123)(92,128)(93,121)(94,126)(95,131)(96,124)(97,107)(98,100)(99,105)(101,103)(102,108)(104,106)>;
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)(133,134,135,136,137,138,139,140,141,142,143,144), (1,136,17,76,126,55,45,65,94,102,25,114)(2,137,18,77,127,56,46,66,95,103,26,115)(3,138,19,78,128,57,47,67,96,104,27,116)(4,139,20,79,129,58,48,68,85,105,28,117)(5,140,21,80,130,59,37,69,86,106,29,118)(6,141,22,81,131,60,38,70,87,107,30,119)(7,142,23,82,132,49,39,71,88,108,31,120)(8,143,24,83,121,50,40,72,89,97,32,109)(9,144,13,84,122,51,41,61,90,98,33,110)(10,133,14,73,123,52,42,62,91,99,34,111)(11,134,15,74,124,53,43,63,92,100,35,112)(12,135,16,75,125,54,44,64,93,101,36,113), (2,6)(3,11)(5,9)(8,12)(13,29)(14,34)(15,27)(16,32)(17,25)(18,30)(19,35)(20,28)(21,33)(22,26)(23,31)(24,36)(37,41)(38,46)(40,44)(43,47)(49,136)(50,141)(51,134)(52,139)(53,144)(54,137)(55,142)(56,135)(57,140)(58,133)(59,138)(60,143)(61,112)(62,117)(63,110)(64,115)(65,120)(66,113)(67,118)(68,111)(69,116)(70,109)(71,114)(72,119)(73,79)(74,84)(75,77)(76,82)(78,80)(81,83)(85,129)(86,122)(87,127)(88,132)(89,125)(90,130)(91,123)(92,128)(93,121)(94,126)(95,131)(96,124)(97,107)(98,100)(99,105)(101,103)(102,108)(104,106) );
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),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,136,17,76,126,55,45,65,94,102,25,114),(2,137,18,77,127,56,46,66,95,103,26,115),(3,138,19,78,128,57,47,67,96,104,27,116),(4,139,20,79,129,58,48,68,85,105,28,117),(5,140,21,80,130,59,37,69,86,106,29,118),(6,141,22,81,131,60,38,70,87,107,30,119),(7,142,23,82,132,49,39,71,88,108,31,120),(8,143,24,83,121,50,40,72,89,97,32,109),(9,144,13,84,122,51,41,61,90,98,33,110),(10,133,14,73,123,52,42,62,91,99,34,111),(11,134,15,74,124,53,43,63,92,100,35,112),(12,135,16,75,125,54,44,64,93,101,36,113)], [(2,6),(3,11),(5,9),(8,12),(13,29),(14,34),(15,27),(16,32),(17,25),(18,30),(19,35),(20,28),(21,33),(22,26),(23,31),(24,36),(37,41),(38,46),(40,44),(43,47),(49,136),(50,141),(51,134),(52,139),(53,144),(54,137),(55,142),(56,135),(57,140),(58,133),(59,138),(60,143),(61,112),(62,117),(63,110),(64,115),(65,120),(66,113),(67,118),(68,111),(69,116),(70,109),(71,114),(72,119),(73,79),(74,84),(75,77),(76,82),(78,80),(81,83),(85,129),(86,122),(87,127),(88,132),(89,125),(90,130),(91,123),(92,128),(93,121),(94,126),(95,131),(96,124),(97,107),(98,100),(99,105),(101,103),(102,108),(104,106)])
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