metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊2D4, D18⋊3D4, C23.13D18, (C2×D4)⋊4D9, (D4×C18)⋊3C2, C9⋊4(C4⋊D4), C4⋊2(C9⋊D4), (C6×D4).8S3, C2.26(D4×D9), C4⋊Dic9⋊14C2, C3.(D6⋊3D4), (C2×C4).52D18, C6.101(S3×D4), C18.51(C2×D4), (C2×C12).60D6, (C22×C6).51D6, C18.31(C4○D4), C12.13(C3⋊D4), (C2×C36).38C22, (C2×C18).53C23, C6.88(D4⋊2S3), C2.17(D4⋊2D9), C18.D4⋊11C2, C22.60(C22×D9), (C22×C18).20C22, (C2×Dic9).17C22, (C22×D9).26C22, (C2×C4×D9)⋊2C2, (C2×C9⋊D4)⋊5C2, C6.98(C2×C3⋊D4), C2.14(C2×C9⋊D4), (C2×C6).210(C22×S3), SmallGroup(288,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊2D4
G = < a,b,c | a36=b4=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >
Subgroups: 612 in 141 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D9, C18, C18, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, Dic9, C36, D18, D18, C2×C18, C2×C18, C4⋊Dic3, C6.D4, S3×C2×C4, C2×C3⋊D4, C6×D4, C4×D9, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×C18, D6⋊3D4, C4⋊Dic9, C18.D4, C2×C4×D9, C2×C9⋊D4, D4×C18, C36⋊2D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C4⋊D4, D18, S3×D4, D4⋊2S3, C2×C3⋊D4, C9⋊D4, C22×D9, D6⋊3D4, D4×D9, D4⋊2D9, C2×C9⋊D4, C36⋊2D4
►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 83 47 119)(2 82 48 118)(3 81 49 117)(4 80 50 116)(5 79 51 115)(6 78 52 114)(7 77 53 113)(8 76 54 112)(9 75 55 111)(10 74 56 110)(11 73 57 109)(12 108 58 144)(13 107 59 143)(14 106 60 142)(15 105 61 141)(16 104 62 140)(17 103 63 139)(18 102 64 138)(19 101 65 137)(20 100 66 136)(21 99 67 135)(22 98 68 134)(23 97 69 133)(24 96 70 132)(25 95 71 131)(26 94 72 130)(27 93 37 129)(28 92 38 128)(29 91 39 127)(30 90 40 126)(31 89 41 125)(32 88 42 124)(33 87 43 123)(34 86 44 122)(35 85 45 121)(36 84 46 120)
(2 18)(3 35)(4 16)(5 33)(6 14)(7 31)(8 12)(9 29)(11 27)(13 25)(15 23)(17 21)(20 36)(22 34)(24 32)(26 30)(37 57)(39 55)(40 72)(41 53)(42 70)(43 51)(44 68)(45 49)(46 66)(48 64)(50 62)(52 60)(54 58)(59 71)(61 69)(63 67)(73 129)(74 110)(75 127)(76 144)(77 125)(78 142)(79 123)(80 140)(81 121)(82 138)(83 119)(84 136)(85 117)(86 134)(87 115)(88 132)(89 113)(90 130)(91 111)(92 128)(93 109)(94 126)(95 143)(96 124)(97 141)(98 122)(99 139)(100 120)(101 137)(102 118)(103 135)(104 116)(105 133)(106 114)(107 131)(108 112)
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), (1,83,47,119)(2,82,48,118)(3,81,49,117)(4,80,50,116)(5,79,51,115)(6,78,52,114)(7,77,53,113)(8,76,54,112)(9,75,55,111)(10,74,56,110)(11,73,57,109)(12,108,58,144)(13,107,59,143)(14,106,60,142)(15,105,61,141)(16,104,62,140)(17,103,63,139)(18,102,64,138)(19,101,65,137)(20,100,66,136)(21,99,67,135)(22,98,68,134)(23,97,69,133)(24,96,70,132)(25,95,71,131)(26,94,72,130)(27,93,37,129)(28,92,38,128)(29,91,39,127)(30,90,40,126)(31,89,41,125)(32,88,42,124)(33,87,43,123)(34,86,44,122)(35,85,45,121)(36,84,46,120), (2,18)(3,35)(4,16)(5,33)(6,14)(7,31)(8,12)(9,29)(11,27)(13,25)(15,23)(17,21)(20,36)(22,34)(24,32)(26,30)(37,57)(39,55)(40,72)(41,53)(42,70)(43,51)(44,68)(45,49)(46,66)(48,64)(50,62)(52,60)(54,58)(59,71)(61,69)(63,67)(73,129)(74,110)(75,127)(76,144)(77,125)(78,142)(79,123)(80,140)(81,121)(82,138)(83,119)(84,136)(85,117)(86,134)(87,115)(88,132)(89,113)(90,130)(91,111)(92,128)(93,109)(94,126)(95,143)(96,124)(97,141)(98,122)(99,139)(100,120)(101,137)(102,118)(103,135)(104,116)(105,133)(106,114)(107,131)(108,112)>;
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), (1,83,47,119)(2,82,48,118)(3,81,49,117)(4,80,50,116)(5,79,51,115)(6,78,52,114)(7,77,53,113)(8,76,54,112)(9,75,55,111)(10,74,56,110)(11,73,57,109)(12,108,58,144)(13,107,59,143)(14,106,60,142)(15,105,61,141)(16,104,62,140)(17,103,63,139)(18,102,64,138)(19,101,65,137)(20,100,66,136)(21,99,67,135)(22,98,68,134)(23,97,69,133)(24,96,70,132)(25,95,71,131)(26,94,72,130)(27,93,37,129)(28,92,38,128)(29,91,39,127)(30,90,40,126)(31,89,41,125)(32,88,42,124)(33,87,43,123)(34,86,44,122)(35,85,45,121)(36,84,46,120), (2,18)(3,35)(4,16)(5,33)(6,14)(7,31)(8,12)(9,29)(11,27)(13,25)(15,23)(17,21)(20,36)(22,34)(24,32)(26,30)(37,57)(39,55)(40,72)(41,53)(42,70)(43,51)(44,68)(45,49)(46,66)(48,64)(50,62)(52,60)(54,58)(59,71)(61,69)(63,67)(73,129)(74,110)(75,127)(76,144)(77,125)(78,142)(79,123)(80,140)(81,121)(82,138)(83,119)(84,136)(85,117)(86,134)(87,115)(88,132)(89,113)(90,130)(91,111)(92,128)(93,109)(94,126)(95,143)(96,124)(97,141)(98,122)(99,139)(100,120)(101,137)(102,118)(103,135)(104,116)(105,133)(106,114)(107,131)(108,112) );
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)], [(1,83,47,119),(2,82,48,118),(3,81,49,117),(4,80,50,116),(5,79,51,115),(6,78,52,114),(7,77,53,113),(8,76,54,112),(9,75,55,111),(10,74,56,110),(11,73,57,109),(12,108,58,144),(13,107,59,143),(14,106,60,142),(15,105,61,141),(16,104,62,140),(17,103,63,139),(18,102,64,138),(19,101,65,137),(20,100,66,136),(21,99,67,135),(22,98,68,134),(23,97,69,133),(24,96,70,132),(25,95,71,131),(26,94,72,130),(27,93,37,129),(28,92,38,128),(29,91,39,127),(30,90,40,126),(31,89,41,125),(32,88,42,124),(33,87,43,123),(34,86,44,122),(35,85,45,121),(36,84,46,120)], [(2,18),(3,35),(4,16),(5,33),(6,14),(7,31),(8,12),(9,29),(11,27),(13,25),(15,23),(17,21),(20,36),(22,34),(24,32),(26,30),(37,57),(39,55),(40,72),(41,53),(42,70),(43,51),(44,68),(45,49),(46,66),(48,64),(50,62),(52,60),(54,58),(59,71),(61,69),(63,67),(73,129),(74,110),(75,127),(76,144),(77,125),(78,142),(79,123),(80,140),(81,121),(82,138),(83,119),(84,136),(85,117),(86,134),(87,115),(88,132),(89,113),(90,130),(91,111),(92,128),(93,109),(94,126),(95,143),(96,124),(97,141),(98,122),(99,139),(100,120),(101,137),(102,118),(103,135),(104,116),(105,133),(106,114),(107,131),(108,112)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | S3×D4 | D4⋊2S3 | D4×D9 | D4⋊2D9 |
| kernel | C36⋊2D4 | C4⋊Dic9 | C18.D4 | C2×C4×D9 | C2×C9⋊D4 | D4×C18 | C6×D4 | C36 | D18 | C2×C12 | C22×C6 | C18 | C2×D4 | C12 | C2×C4 | C23 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 3 | 4 | 3 | 6 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of C36⋊2D4 ►in GL6(𝔽37)
| 17 | 11 | 0 | 0 | 0 | 0 |
| 26 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
G:=sub<GL(6,GF(37))| [17,26,0,0,0,0,11,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,0,0,0,6],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0],[1,36,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,36] >;
C36⋊2D4 in GAP, Magma, Sage, TeX
C_{36}\rtimes_2D_4 % in TeX
G:=Group("C36:2D4"); // GroupNames label
G:=SmallGroup(288,148);
// by ID
G=gap.SmallGroup(288,148);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations