metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D36⋊5C4, Dic9⋊5D4, C9⋊3(C4×D4), C4⋊C4⋊8D9, C4⋊1(C4×D9), C36⋊2(C2×C4), C2.4(D4×D9), D18⋊3(C2×C4), C6.85(S3×D4), D18⋊C4⋊12C2, C12.10(C4×S3), (C4×Dic9)⋊3C2, (C2×D36).7C2, C18.24(C2×D4), (C2×C4).32D18, C3.(Dic3⋊5D4), (C2×C12).182D6, C18.33(C4○D4), (C2×C36).57C22, (C2×C18).34C23, C18.11(C22×C4), C2.2(Q8⋊3D9), C6.39(Q8⋊3S3), C22.18(C22×D9), (C2×Dic9).33C22, (C22×D9).19C22, (C9×C4⋊C4)⋊4C2, (C2×C4×D9)⋊12C2, C6.50(S3×C2×C4), C2.13(C2×C4×D9), (C3×C4⋊C4).11S3, (C2×C6).191(C22×S3), SmallGroup(288,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36⋊C4
G = < a,b,c | a36=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a18b >
Subgroups: 656 in 141 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×D4, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C4×D9, D36, C2×Dic9, C2×C36, C2×C36, C22×D9, Dic3⋊5D4, C4×Dic9, D18⋊C4, C9×C4⋊C4, C2×C4×D9, C2×D36, D36⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, D9, C4×S3, C22×S3, C4×D4, D18, S3×C2×C4, S3×D4, Q8⋊3S3, C4×D9, C22×D9, Dic3⋊5D4, C2×C4×D9, D4×D9, Q8⋊3D9, D36⋊C4
►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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)(37 43)(38 42)(39 41)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)
(1 115 102 72)(2 134 103 55)(3 117 104 38)(4 136 105 57)(5 119 106 40)(6 138 107 59)(7 121 108 42)(8 140 73 61)(9 123 74 44)(10 142 75 63)(11 125 76 46)(12 144 77 65)(13 127 78 48)(14 110 79 67)(15 129 80 50)(16 112 81 69)(17 131 82 52)(18 114 83 71)(19 133 84 54)(20 116 85 37)(21 135 86 56)(22 118 87 39)(23 137 88 58)(24 120 89 41)(25 139 90 60)(26 122 91 43)(27 141 92 62)(28 124 93 45)(29 143 94 64)(30 126 95 47)(31 109 96 66)(32 128 97 49)(33 111 98 68)(34 130 99 51)(35 113 100 70)(36 132 101 53)
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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)(37,43)(38,42)(39,41)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138), (1,115,102,72)(2,134,103,55)(3,117,104,38)(4,136,105,57)(5,119,106,40)(6,138,107,59)(7,121,108,42)(8,140,73,61)(9,123,74,44)(10,142,75,63)(11,125,76,46)(12,144,77,65)(13,127,78,48)(14,110,79,67)(15,129,80,50)(16,112,81,69)(17,131,82,52)(18,114,83,71)(19,133,84,54)(20,116,85,37)(21,135,86,56)(22,118,87,39)(23,137,88,58)(24,120,89,41)(25,139,90,60)(26,122,91,43)(27,141,92,62)(28,124,93,45)(29,143,94,64)(30,126,95,47)(31,109,96,66)(32,128,97,49)(33,111,98,68)(34,130,99,51)(35,113,100,70)(36,132,101,53)>;
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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)(37,43)(38,42)(39,41)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138), (1,115,102,72)(2,134,103,55)(3,117,104,38)(4,136,105,57)(5,119,106,40)(6,138,107,59)(7,121,108,42)(8,140,73,61)(9,123,74,44)(10,142,75,63)(11,125,76,46)(12,144,77,65)(13,127,78,48)(14,110,79,67)(15,129,80,50)(16,112,81,69)(17,131,82,52)(18,114,83,71)(19,133,84,54)(20,116,85,37)(21,135,86,56)(22,118,87,39)(23,137,88,58)(24,120,89,41)(25,139,90,60)(26,122,91,43)(27,141,92,62)(28,124,93,45)(29,143,94,64)(30,126,95,47)(31,109,96,66)(32,128,97,49)(33,111,98,68)(34,130,99,51)(35,113,100,70)(36,132,101,53) );
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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(28,36),(29,35),(30,34),(31,33),(37,43),(38,42),(39,41),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(109,129),(110,128),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(130,144),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138)], [(1,115,102,72),(2,134,103,55),(3,117,104,38),(4,136,105,57),(5,119,106,40),(6,138,107,59),(7,121,108,42),(8,140,73,61),(9,123,74,44),(10,142,75,63),(11,125,76,46),(12,144,77,65),(13,127,78,48),(14,110,79,67),(15,129,80,50),(16,112,81,69),(17,131,82,52),(18,114,83,71),(19,133,84,54),(20,116,85,37),(21,135,86,56),(22,118,87,39),(23,137,88,58),(24,120,89,41),(25,139,90,60),(26,122,91,43),(27,141,92,62),(28,124,93,45),(29,143,94,64),(30,126,95,47),(31,109,96,66),(32,128,97,49),(33,111,98,68),(34,130,99,51),(35,113,100,70),(36,132,101,53)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | D9 | C4×S3 | D18 | C4×D9 | S3×D4 | Q8⋊3S3 | D4×D9 | Q8⋊3D9 |
| kernel | D36⋊C4 | C4×Dic9 | D18⋊C4 | C9×C4⋊C4 | C2×C4×D9 | C2×D36 | D36 | C3×C4⋊C4 | Dic9 | C2×C12 | C18 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 3 | 2 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of D36⋊C4 ►in GL4(𝔽37) generated by
| 31 | 11 | 0 | 0 |
| 26 | 20 | 0 | 0 |
| 0 | 0 | 36 | 2 |
| 0 | 0 | 36 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 36 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 1 | 35 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [31,26,0,0,11,20,0,0,0,0,36,36,0,0,2,1],[1,0,0,0,1,36,0,0,0,0,36,36,0,0,0,1],[6,0,0,0,0,6,0,0,0,0,1,0,0,0,35,36] >;
D36⋊C4 in GAP, Magma, Sage, TeX
D_{36}\rtimes C_4 % in TeX
G:=Group("D36:C4"); // GroupNames label
G:=SmallGroup(288,103);
// by ID
G=gap.SmallGroup(288,103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations