direct product, metabelian, supersoluble, monomial
Aliases: C2×C12.59D6, C62.277C23, (C2×C12)⋊29D6, C6⋊5(C4○D12), (C22×C12)⋊12S3, (C6×C12)⋊32C22, C6.57(S3×C23), (C3×C6).56C24, (C22×C6).167D6, C12⋊S3⋊29C22, C12.214(C22×S3), (C3×C12).184C23, C32⋊7D4⋊16C22, C3⋊Dic3.46C23, C32⋊4Q8⋊27C22, (C2×C62).123C22, (C2×C6×C12)⋊10C2, C3⋊6(C2×C4○D12), (C3×C6)⋊8(C4○D4), C32⋊14(C2×C4○D4), C2.5(C23×C3⋊S3), (C4×C3⋊S3)⋊20C22, (C22×C4)⋊8(C3⋊S3), (C2×C12⋊S3)⋊23C2, C23.32(C2×C3⋊S3), C4.43(C22×C3⋊S3), (C2×C3⋊S3).50C23, (C2×C32⋊7D4)⋊21C2, (C2×C32⋊4Q8)⋊24C2, C22.5(C22×C3⋊S3), (C2×C6).286(C22×S3), (C22×C3⋊S3).106C22, (C2×C3⋊Dic3).182C22, (C2×C4×C3⋊S3)⋊26C2, (C2×C4)⋊10(C2×C3⋊S3), SmallGroup(288,1006)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1636 in 492 conjugacy classes, 165 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×4], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×16], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×16], C12 [×16], D6 [×32], C2×C6 [×12], C2×C6 [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×16], C4×S3 [×32], D12 [×16], C2×Dic3 [×8], C3⋊D4 [×32], C2×C12 [×24], C22×S3 [×8], C22×C6 [×4], C2×C4○D4, C3⋊Dic3 [×4], C3×C12 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C62 [×2], C62 [×2], C2×Dic6 [×4], S3×C2×C4 [×8], C2×D12 [×4], C4○D12 [×32], C2×C3⋊D4 [×8], C22×C12 [×4], C32⋊4Q8 [×4], C4×C3⋊S3 [×8], C12⋊S3 [×4], C2×C3⋊Dic3 [×2], C32⋊7D4 [×8], C6×C12 [×2], C6×C12 [×4], C22×C3⋊S3 [×2], C2×C62, C2×C4○D12 [×4], C2×C32⋊4Q8, C2×C4×C3⋊S3 [×2], C2×C12⋊S3, C12.59D6 [×8], C2×C32⋊7D4 [×2], C2×C6×C12, C2×C12.59D6
Quotients:
C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C4○D4 [×2], C24, C3⋊S3, C22×S3 [×28], C2×C4○D4, C2×C3⋊S3 [×7], C4○D12 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×C4○D12 [×4], C12.59D6 [×2], C23×C3⋊S3, C2×C12.59D6
Generators and relations
G = < a,b,c,d | a2=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >
►On 144 points
Generators in S144
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(73 107)(74 108)(75 97)(76 98)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 106)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 133)(93 134)(94 135)(95 136)(96 137)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 121)(120 122)
(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)
(1 106 63 85 49 114)(2 107 64 86 50 115)(3 108 65 87 51 116)(4 97 66 88 52 117)(5 98 67 89 53 118)(6 99 68 90 54 119)(7 100 69 91 55 120)(8 101 70 92 56 109)(9 102 71 93 57 110)(10 103 72 94 58 111)(11 104 61 95 59 112)(12 105 62 96 60 113)(13 142 41 132 31 76)(14 143 42 121 32 77)(15 144 43 122 33 78)(16 133 44 123 34 79)(17 134 45 124 35 80)(18 135 46 125 36 81)(19 136 47 126 25 82)(20 137 48 127 26 83)(21 138 37 128 27 84)(22 139 38 129 28 73)(23 140 39 130 29 74)(24 141 40 131 30 75)
(1 128 7 122)(2 121 8 127)(3 126 9 132)(4 131 10 125)(5 124 11 130)(6 129 12 123)(13 93 19 87)(14 86 20 92)(15 91 21 85)(16 96 22 90)(17 89 23 95)(18 94 24 88)(25 116 31 110)(26 109 32 115)(27 114 33 120)(28 119 34 113)(29 112 35 118)(30 117 36 111)(37 106 43 100)(38 99 44 105)(39 104 45 98)(40 97 46 103)(41 102 47 108)(42 107 48 101)(49 84 55 78)(50 77 56 83)(51 82 57 76)(52 75 58 81)(53 80 59 74)(54 73 60 79)(61 140 67 134)(62 133 68 139)(63 138 69 144)(64 143 70 137)(65 136 71 142)(66 141 72 135)
G:=sub<Sym(144)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(73,107)(74,108)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,106)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,133)(93,134)(94,135)(95,136)(96,137)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,121)(120,122), (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), (1,106,63,85,49,114)(2,107,64,86,50,115)(3,108,65,87,51,116)(4,97,66,88,52,117)(5,98,67,89,53,118)(6,99,68,90,54,119)(7,100,69,91,55,120)(8,101,70,92,56,109)(9,102,71,93,57,110)(10,103,72,94,58,111)(11,104,61,95,59,112)(12,105,62,96,60,113)(13,142,41,132,31,76)(14,143,42,121,32,77)(15,144,43,122,33,78)(16,133,44,123,34,79)(17,134,45,124,35,80)(18,135,46,125,36,81)(19,136,47,126,25,82)(20,137,48,127,26,83)(21,138,37,128,27,84)(22,139,38,129,28,73)(23,140,39,130,29,74)(24,141,40,131,30,75), (1,128,7,122)(2,121,8,127)(3,126,9,132)(4,131,10,125)(5,124,11,130)(6,129,12,123)(13,93,19,87)(14,86,20,92)(15,91,21,85)(16,96,22,90)(17,89,23,95)(18,94,24,88)(25,116,31,110)(26,109,32,115)(27,114,33,120)(28,119,34,113)(29,112,35,118)(30,117,36,111)(37,106,43,100)(38,99,44,105)(39,104,45,98)(40,97,46,103)(41,102,47,108)(42,107,48,101)(49,84,55,78)(50,77,56,83)(51,82,57,76)(52,75,58,81)(53,80,59,74)(54,73,60,79)(61,140,67,134)(62,133,68,139)(63,138,69,144)(64,143,70,137)(65,136,71,142)(66,141,72,135)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(73,107)(74,108)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,106)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,133)(93,134)(94,135)(95,136)(96,137)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,121)(120,122), (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), (1,106,63,85,49,114)(2,107,64,86,50,115)(3,108,65,87,51,116)(4,97,66,88,52,117)(5,98,67,89,53,118)(6,99,68,90,54,119)(7,100,69,91,55,120)(8,101,70,92,56,109)(9,102,71,93,57,110)(10,103,72,94,58,111)(11,104,61,95,59,112)(12,105,62,96,60,113)(13,142,41,132,31,76)(14,143,42,121,32,77)(15,144,43,122,33,78)(16,133,44,123,34,79)(17,134,45,124,35,80)(18,135,46,125,36,81)(19,136,47,126,25,82)(20,137,48,127,26,83)(21,138,37,128,27,84)(22,139,38,129,28,73)(23,140,39,130,29,74)(24,141,40,131,30,75), (1,128,7,122)(2,121,8,127)(3,126,9,132)(4,131,10,125)(5,124,11,130)(6,129,12,123)(13,93,19,87)(14,86,20,92)(15,91,21,85)(16,96,22,90)(17,89,23,95)(18,94,24,88)(25,116,31,110)(26,109,32,115)(27,114,33,120)(28,119,34,113)(29,112,35,118)(30,117,36,111)(37,106,43,100)(38,99,44,105)(39,104,45,98)(40,97,46,103)(41,102,47,108)(42,107,48,101)(49,84,55,78)(50,77,56,83)(51,82,57,76)(52,75,58,81)(53,80,59,74)(54,73,60,79)(61,140,67,134)(62,133,68,139)(63,138,69,144)(64,143,70,137)(65,136,71,142)(66,141,72,135) );
G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,25),(12,26),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60),(73,107),(74,108),(75,97),(76,98),(77,99),(78,100),(79,101),(80,102),(81,103),(82,104),(83,105),(84,106),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143),(91,144),(92,133),(93,134),(94,135),(95,136),(96,137),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,121),(120,122)], [(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)], [(1,106,63,85,49,114),(2,107,64,86,50,115),(3,108,65,87,51,116),(4,97,66,88,52,117),(5,98,67,89,53,118),(6,99,68,90,54,119),(7,100,69,91,55,120),(8,101,70,92,56,109),(9,102,71,93,57,110),(10,103,72,94,58,111),(11,104,61,95,59,112),(12,105,62,96,60,113),(13,142,41,132,31,76),(14,143,42,121,32,77),(15,144,43,122,33,78),(16,133,44,123,34,79),(17,134,45,124,35,80),(18,135,46,125,36,81),(19,136,47,126,25,82),(20,137,48,127,26,83),(21,138,37,128,27,84),(22,139,38,129,28,73),(23,140,39,130,29,74),(24,141,40,131,30,75)], [(1,128,7,122),(2,121,8,127),(3,126,9,132),(4,131,10,125),(5,124,11,130),(6,129,12,123),(13,93,19,87),(14,86,20,92),(15,91,21,85),(16,96,22,90),(17,89,23,95),(18,94,24,88),(25,116,31,110),(26,109,32,115),(27,114,33,120),(28,119,34,113),(29,112,35,118),(30,117,36,111),(37,106,43,100),(38,99,44,105),(39,104,45,98),(40,97,46,103),(41,102,47,108),(42,107,48,101),(49,84,55,78),(50,77,56,83),(51,82,57,76),(52,75,58,81),(53,80,59,74),(54,73,60,79),(61,140,67,134),(62,133,68,139),(63,138,69,144),(64,143,70,137),(65,136,71,142),(66,141,72,135)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 5 | 0 |
| 1 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 2 | 2 |
| 12 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 11 | 4 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,5,5,0,0,8,0],[1,12,0,0,1,0,0,0,0,0,4,2,0,0,11,2],[12,0,0,0,12,1,0,0,0,0,9,11,0,0,2,4] >;
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 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 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C2×C12.59D6 | C2×C32⋊4Q8 | C2×C4×C3⋊S3 | C2×C12⋊S3 | C12.59D6 | C2×C32⋊7D4 | C2×C6×C12 | C22×C12 | C2×C12 | C22×C6 | C3×C6 | C6 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 4 | 24 | 4 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2\times C_{12}._{59}D_6 % in TeX
G:=Group("C2xC12.59D6"); // GroupNames label
G:=SmallGroup(288,1006);
// by ID
G=gap.SmallGroup(288,1006);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀