direct product, metabelian, supersoluble, monomial
Aliases: C2×C36.C6, Dic18⋊5C6, C62.37D6, C18⋊(C3×Q8), C9⋊1(C6×Q8), (C2×C36).3C6, (C2×Dic18)⋊C3, C12.78(S3×C6), C36.11(C2×C6), (C6×C12).13S3, (C3×C12).51D6, C3.3(C6×Dic6), C32.(C2×Dic6), C9⋊C12.1C22, C18.1(C22×C6), (C2×Dic9).3C6, Dic9.1(C2×C6), C6.11(C3×Dic6), (C3×C6).11Dic6, (C2×3- 1+2)⋊Q8, 3- 1+2⋊1(C2×Q8), (C2×3- 1+2).1C23, (C4×3- 1+2).11C22, (C22×3- 1+2).7C22, C6.27(S3×C2×C6), C4.11(C2×C9⋊C6), (C2×C9⋊C12).3C2, (C2×C4).4(C9⋊C6), (C2×C18).7(C2×C6), (C2×C6).57(S3×C6), C2.3(C22×C9⋊C6), C22.8(C2×C9⋊C6), (C2×C12).18(C3×S3), (C3×C6).23(C22×S3), (C2×C4×3- 1+2).3C2, SmallGroup(432,352)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C36.C6
G = < a,b,c | a2=b36=1, c6=b18, ab=ba, ac=ca, cbc-1=b11 >
Subgroups: 366 in 122 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, 3- 1+2, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C2×3- 1+2, C2×3- 1+2, Dic18, C2×Dic9, C2×C36, C2×C36, C3×Dic6, C6×Dic3, C6×C12, C9⋊C12, C4×3- 1+2, C22×3- 1+2, C2×Dic18, C6×Dic6, C36.C6, C2×C9⋊C12, C2×C4×3- 1+2, C2×C36.C6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C2×Dic6, C6×Q8, C9⋊C6, C3×Dic6, S3×C2×C6, C2×C9⋊C6, C6×Dic6, C36.C6, C22×C9⋊C6, C2×C36.C6
►On 144 points
Generators in S144
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)
(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)
(1 117 19 135)(2 140 8 134 14 128 20 122 26 116 32 110)(3 127 33 133 27 139 21 109 15 115 9 121)(4 114 22 132)(5 137 11 131 17 125 23 119 29 113 35 143)(6 124 36 130 30 136 24 142 18 112 12 118)(7 111 25 129)(10 144 28 126)(13 141 31 123)(16 138 34 120)(37 101 43 95 49 89 55 83 61 77 67 107)(38 88 68 94 62 100 56 106 50 76 44 82)(39 75 57 93)(40 98 46 92 52 86 58 80 64 74 70 104)(41 85 71 91 65 97 59 103 53 73 47 79)(42 108 60 90)(45 105 63 87)(48 102 66 84)(51 99 69 81)(54 96 72 78)
G:=sub<Sym(144)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120), (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), (1,117,19,135)(2,140,8,134,14,128,20,122,26,116,32,110)(3,127,33,133,27,139,21,109,15,115,9,121)(4,114,22,132)(5,137,11,131,17,125,23,119,29,113,35,143)(6,124,36,130,30,136,24,142,18,112,12,118)(7,111,25,129)(10,144,28,126)(13,141,31,123)(16,138,34,120)(37,101,43,95,49,89,55,83,61,77,67,107)(38,88,68,94,62,100,56,106,50,76,44,82)(39,75,57,93)(40,98,46,92,52,86,58,80,64,74,70,104)(41,85,71,91,65,97,59,103,53,73,47,79)(42,108,60,90)(45,105,63,87)(48,102,66,84)(51,99,69,81)(54,96,72,78)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120), (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), (1,117,19,135)(2,140,8,134,14,128,20,122,26,116,32,110)(3,127,33,133,27,139,21,109,15,115,9,121)(4,114,22,132)(5,137,11,131,17,125,23,119,29,113,35,143)(6,124,36,130,30,136,24,142,18,112,12,118)(7,111,25,129)(10,144,28,126)(13,141,31,123)(16,138,34,120)(37,101,43,95,49,89,55,83,61,77,67,107)(38,88,68,94,62,100,56,106,50,76,44,82)(39,75,57,93)(40,98,46,92,52,86,58,80,64,74,70,104)(41,85,71,91,65,97,59,103,53,73,47,79)(42,108,60,90)(45,105,63,87)(48,102,66,84)(51,99,69,81)(54,96,72,78) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(97,109),(98,110),(99,111),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,118),(107,119),(108,120)], [(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)], [(1,117,19,135),(2,140,8,134,14,128,20,122,26,116,32,110),(3,127,33,133,27,139,21,109,15,115,9,121),(4,114,22,132),(5,137,11,131,17,125,23,119,29,113,35,143),(6,124,36,130,30,136,24,142,18,112,12,118),(7,111,25,129),(10,144,28,126),(13,141,31,123),(16,138,34,120),(37,101,43,95,49,89,55,83,61,77,67,107),(38,88,68,94,62,100,56,106,50,76,44,82),(39,75,57,93),(40,98,46,92,52,86,58,80,64,74,70,104),(41,85,71,91,65,97,59,103,53,73,47,79),(42,108,60,90),(45,105,63,87),(48,102,66,84),(51,99,69,81),(54,96,72,78)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | ··· | 18 | 6 | ··· | 6 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | + | + | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | D6 | C3×S3 | C3×Q8 | Dic6 | S3×C6 | S3×C6 | C3×Dic6 | C9⋊C6 | C2×C9⋊C6 | C2×C9⋊C6 | C36.C6 |
| kernel | C2×C36.C6 | C36.C6 | C2×C9⋊C12 | C2×C4×3- 1+2 | C2×Dic18 | Dic18 | C2×Dic9 | C2×C36 | C6×C12 | C2×3- 1+2 | C3×C12 | C62 | C2×C12 | C18 | C3×C6 | C12 | C2×C6 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 4 |
Matrix representation of C2×C36.C6 ►in GL8(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 36 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 25 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 29 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 0 | 12 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 25 | 0 | 0 |
G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36],[36,24,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0],[25,4,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,29,12,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,0,0,12,33,0,0,0,0,0,0,8,25,0,0,0,0,29,12,0,0,0,0,0,0,4,8,0,0] >;
C2×C36.C6 in GAP, Magma, Sage, TeX
C_2\times C_{36}.C_6 % in TeX
G:=Group("C2xC36.C6"); // GroupNames label
G:=SmallGroup(432,352);
// by ID
G=gap.SmallGroup(432,352);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,10085,1034,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^2=b^36=1,c^6=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations