metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊3D4, Dic9⋊1D4, C23.15D18, (C2×D4)⋊6D9, (D4×C18)⋊4C2, (C2×D36)⋊9C2, C4⋊1(C9⋊D4), C9⋊2(C4⋊1D4), (C6×D4).9S3, C2.28(D4×D9), C3.(C12⋊3D4), (C4×Dic9)⋊6C2, (C2×C4).53D18, (C2×C12).61D6, C6.103(S3×D4), C18.52(C2×D4), (C22×C6).53D6, C12.14(C3⋊D4), (C2×C18).55C23, (C2×C36).39C22, C22.62(C22×D9), (C22×C18).22C22, (C2×Dic9).41C22, (C22×D9).12C22, (C2×C9⋊D4)⋊7C2, C2.16(C2×C9⋊D4), C6.100(C2×C3⋊D4), (C2×C6).212(C22×S3), SmallGroup(288,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊D4
G = < a,b,c | a36=b4=c2=1, bab-1=a17, cac=a-1, cbc=b-1 >
Subgroups: 820 in 162 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×D4, C2×D4, D9, C18, C18, C18, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊1D4, Dic9, C36, D18, C2×C18, C2×C18, C4×Dic3, C2×D12, C2×C3⋊D4, C6×D4, D36, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×C18, C12⋊3D4, C4×Dic9, C2×D36, C2×C9⋊D4, D4×C18, C36⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C4⋊1D4, D18, S3×D4, C2×C3⋊D4, C9⋊D4, C22×D9, C12⋊3D4, D4×D9, C2×C9⋊D4, C36⋊D4
►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)
(1 111 43 96)(2 128 44 77)(3 109 45 94)(4 126 46 75)(5 143 47 92)(6 124 48 73)(7 141 49 90)(8 122 50 107)(9 139 51 88)(10 120 52 105)(11 137 53 86)(12 118 54 103)(13 135 55 84)(14 116 56 101)(15 133 57 82)(16 114 58 99)(17 131 59 80)(18 112 60 97)(19 129 61 78)(20 110 62 95)(21 127 63 76)(22 144 64 93)(23 125 65 74)(24 142 66 91)(25 123 67 108)(26 140 68 89)(27 121 69 106)(28 138 70 87)(29 119 71 104)(30 136 72 85)(31 117 37 102)(32 134 38 83)(33 115 39 100)(34 132 40 81)(35 113 41 98)(36 130 42 79)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(73 134)(74 133)(75 132)(76 131)(77 130)(78 129)(79 128)(80 127)(81 126)(82 125)(83 124)(84 123)(85 122)(86 121)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 144)(100 143)(101 142)(102 141)(103 140)(104 139)(105 138)(106 137)(107 136)(108 135)
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), (1,111,43,96)(2,128,44,77)(3,109,45,94)(4,126,46,75)(5,143,47,92)(6,124,48,73)(7,141,49,90)(8,122,50,107)(9,139,51,88)(10,120,52,105)(11,137,53,86)(12,118,54,103)(13,135,55,84)(14,116,56,101)(15,133,57,82)(16,114,58,99)(17,131,59,80)(18,112,60,97)(19,129,61,78)(20,110,62,95)(21,127,63,76)(22,144,64,93)(23,125,65,74)(24,142,66,91)(25,123,67,108)(26,140,68,89)(27,121,69,106)(28,138,70,87)(29,119,71,104)(30,136,72,85)(31,117,37,102)(32,134,38,83)(33,115,39,100)(34,132,40,81)(35,113,41,98)(36,130,42,79), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)>;
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), (1,111,43,96)(2,128,44,77)(3,109,45,94)(4,126,46,75)(5,143,47,92)(6,124,48,73)(7,141,49,90)(8,122,50,107)(9,139,51,88)(10,120,52,105)(11,137,53,86)(12,118,54,103)(13,135,55,84)(14,116,56,101)(15,133,57,82)(16,114,58,99)(17,131,59,80)(18,112,60,97)(19,129,61,78)(20,110,62,95)(21,127,63,76)(22,144,64,93)(23,125,65,74)(24,142,66,91)(25,123,67,108)(26,140,68,89)(27,121,69,106)(28,138,70,87)(29,119,71,104)(30,136,72,85)(31,117,37,102)(32,134,38,83)(33,115,39,100)(34,132,40,81)(35,113,41,98)(36,130,42,79), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135) );
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)], [(1,111,43,96),(2,128,44,77),(3,109,45,94),(4,126,46,75),(5,143,47,92),(6,124,48,73),(7,141,49,90),(8,122,50,107),(9,139,51,88),(10,120,52,105),(11,137,53,86),(12,118,54,103),(13,135,55,84),(14,116,56,101),(15,133,57,82),(16,114,58,99),(17,131,59,80),(18,112,60,97),(19,129,61,78),(20,110,62,95),(21,127,63,76),(22,144,64,93),(23,125,65,74),(24,142,66,91),(25,123,67,108),(26,140,68,89),(27,121,69,106),(28,138,70,87),(29,119,71,104),(30,136,72,85),(31,117,37,102),(32,134,38,83),(33,115,39,100),(34,132,40,81),(35,113,41,98),(36,130,42,79)], [(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(73,134),(74,133),(75,132),(76,131),(77,130),(78,129),(79,128),(80,127),(81,126),(82,125),(83,124),(84,123),(85,122),(86,121),(87,120),(88,119),(89,118),(90,117),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,110),(98,109),(99,144),(100,143),(101,142),(102,141),(103,140),(104,139),(105,138),(106,137),(107,136),(108,135)]])
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 | 36 | 36 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | S3×D4 | D4×D9 |
| kernel | C36⋊D4 | C4×Dic9 | C2×D36 | C2×C9⋊D4 | D4×C18 | C6×D4 | Dic9 | C36 | C2×C12 | C22×C6 | C2×D4 | C12 | C2×C4 | C23 | C4 | C6 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 2 | 6 |
Matrix representation of C36⋊D4 ►in GL4(𝔽37) generated by
| 17 | 11 | 0 | 0 |
| 26 | 6 | 0 | 0 |
| 0 | 0 | 20 | 1 |
| 0 | 0 | 6 | 17 |
| 2 | 13 | 0 | 0 |
| 11 | 35 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 20 | 26 | 0 | 0 |
| 6 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 36 |
G:=sub<GL(4,GF(37))| [17,26,0,0,11,6,0,0,0,0,20,6,0,0,1,17],[2,11,0,0,13,35,0,0,0,0,36,0,0,0,0,36],[20,6,0,0,26,17,0,0,0,0,1,34,0,0,0,36] >;
C36⋊D4 in GAP, Magma, Sage, TeX
C_{36}\rtimes D_4 % in TeX
G:=Group("C36:D4"); // GroupNames label
G:=SmallGroup(288,150);
// by ID
G=gap.SmallGroup(288,150);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations