Aliases: 2- 1+4⋊C9, C4○D4⋊C18, (C2×Q8)⋊C18, (C3×D4).2A4, D4.(C3.A4), C3.(D4.A4), (C6×Q8).3C6, C12.10(C2×A4), Q8.C18⋊4C2, Q8.3(C2×C18), Q8⋊C9.5C22, C6.18(C22×A4), (C3×2- 1+4).C3, (C2×Q8⋊C9)⋊1C2, C4.3(C2×C3.A4), (C2×C6).15(C2×A4), (C3×C4○D4).4C6, (C3×Q8).14(C2×C6), C2.7(C22×C3.A4), C22.5(C2×C3.A4), SmallGroup(288,349)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — 2- 1+4⋊C9 |
Generators and relations for 2- 1+4⋊C9
G = < a,b,c,d,e | a4=b2=e9=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece-1=d, ede-1=cd >
Subgroups: 201 in 79 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, D4, D4, Q8, Q8, C9, C12, C12, C2×C6, C2×C6, C2×Q8, C2×Q8, C4○D4, C4○D4, C18, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C36, C2×C18, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, Q8⋊C9, D4×C9, C3×2- 1+4, C2×Q8⋊C9, Q8.C18, 2- 1+4⋊C9
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C22×A4, C2×C3.A4, D4.A4, C22×C3.A4, 2- 1+4⋊C9
►On 144 points
Generators in S144
(1 98 85 110)(2 99 86 111)(3 91 87 112)(4 92 88 113)(5 93 89 114)(6 94 90 115)(7 95 82 116)(8 96 83 117)(9 97 84 109)(10 23 70 58)(11 24 71 59)(12 25 72 60)(13 26 64 61)(14 27 65 62)(15 19 66 63)(16 20 67 55)(17 21 68 56)(18 22 69 57)(28 49 136 37)(29 50 137 38)(30 51 138 39)(31 52 139 40)(32 53 140 41)(33 54 141 42)(34 46 142 43)(35 47 143 44)(36 48 144 45)(73 126 127 107)(74 118 128 108)(75 119 129 100)(76 120 130 101)(77 121 131 102)(78 122 132 103)(79 123 133 104)(80 124 134 105)(81 125 135 106)
(1 51)(2 52)(3 53)(4 54)(5 46)(6 47)(7 48)(8 49)(9 50)(10 135)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 118)(26 119)(27 120)(28 96)(29 97)(30 98)(31 99)(32 91)(33 92)(34 93)(35 94)(36 95)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 82)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 100)(62 101)(63 102)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 73)(72 74)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 136)
(1 129 85 75)(2 111 86 99)(3 102 87 121)(4 132 88 78)(5 114 89 93)(6 105 90 124)(7 135 82 81)(8 117 83 96)(9 108 84 118)(10 45 70 48)(11 59 71 24)(12 29 72 137)(13 39 64 51)(14 62 65 27)(15 32 66 140)(16 42 67 54)(17 56 68 21)(18 35 69 143)(19 53 63 41)(20 33 55 141)(22 47 57 44)(23 36 58 144)(25 50 60 38)(26 30 61 138)(28 49 136 37)(31 52 139 40)(34 46 142 43)(73 126 127 107)(74 109 128 97)(76 120 130 101)(77 112 131 91)(79 123 133 104)(80 115 134 94)(92 103 113 122)(95 106 116 125)(98 100 110 119)
(1 110 85 98)(2 101 86 120)(3 131 87 77)(4 113 88 92)(5 104 89 123)(6 134 90 80)(7 116 82 95)(8 107 83 126)(9 128 84 74)(10 58 70 23)(11 28 71 136)(12 38 72 50)(13 61 64 26)(14 31 65 139)(15 41 66 53)(16 55 67 20)(17 34 68 142)(18 44 69 47)(19 32 63 140)(21 46 56 43)(22 35 57 143)(24 49 59 37)(25 29 60 137)(27 52 62 40)(30 51 138 39)(33 54 141 42)(36 48 144 45)(73 117 127 96)(75 119 129 100)(76 111 130 99)(78 122 132 103)(79 114 133 93)(81 125 135 106)(91 102 112 121)(94 105 115 124)(97 108 109 118)
(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)
G:=sub<Sym(144)| (1,98,85,110)(2,99,86,111)(3,91,87,112)(4,92,88,113)(5,93,89,114)(6,94,90,115)(7,95,82,116)(8,96,83,117)(9,97,84,109)(10,23,70,58)(11,24,71,59)(12,25,72,60)(13,26,64,61)(14,27,65,62)(15,19,66,63)(16,20,67,55)(17,21,68,56)(18,22,69,57)(28,49,136,37)(29,50,137,38)(30,51,138,39)(31,52,139,40)(32,53,140,41)(33,54,141,42)(34,46,142,43)(35,47,143,44)(36,48,144,45)(73,126,127,107)(74,118,128,108)(75,119,129,100)(76,120,130,101)(77,121,131,102)(78,122,132,103)(79,123,133,104)(80,124,134,105)(81,125,135,106), (1,51)(2,52)(3,53)(4,54)(5,46)(6,47)(7,48)(8,49)(9,50)(10,135)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,118)(26,119)(27,120)(28,96)(29,97)(30,98)(31,99)(32,91)(33,92)(34,93)(35,94)(36,95)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,82)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,100)(62,101)(63,102)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(109,137)(110,138)(111,139)(112,140)(113,141)(114,142)(115,143)(116,144)(117,136), (1,129,85,75)(2,111,86,99)(3,102,87,121)(4,132,88,78)(5,114,89,93)(6,105,90,124)(7,135,82,81)(8,117,83,96)(9,108,84,118)(10,45,70,48)(11,59,71,24)(12,29,72,137)(13,39,64,51)(14,62,65,27)(15,32,66,140)(16,42,67,54)(17,56,68,21)(18,35,69,143)(19,53,63,41)(20,33,55,141)(22,47,57,44)(23,36,58,144)(25,50,60,38)(26,30,61,138)(28,49,136,37)(31,52,139,40)(34,46,142,43)(73,126,127,107)(74,109,128,97)(76,120,130,101)(77,112,131,91)(79,123,133,104)(80,115,134,94)(92,103,113,122)(95,106,116,125)(98,100,110,119), (1,110,85,98)(2,101,86,120)(3,131,87,77)(4,113,88,92)(5,104,89,123)(6,134,90,80)(7,116,82,95)(8,107,83,126)(9,128,84,74)(10,58,70,23)(11,28,71,136)(12,38,72,50)(13,61,64,26)(14,31,65,139)(15,41,66,53)(16,55,67,20)(17,34,68,142)(18,44,69,47)(19,32,63,140)(21,46,56,43)(22,35,57,143)(24,49,59,37)(25,29,60,137)(27,52,62,40)(30,51,138,39)(33,54,141,42)(36,48,144,45)(73,117,127,96)(75,119,129,100)(76,111,130,99)(78,122,132,103)(79,114,133,93)(81,125,135,106)(91,102,112,121)(94,105,115,124)(97,108,109,118), (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)>;
G:=Group( (1,98,85,110)(2,99,86,111)(3,91,87,112)(4,92,88,113)(5,93,89,114)(6,94,90,115)(7,95,82,116)(8,96,83,117)(9,97,84,109)(10,23,70,58)(11,24,71,59)(12,25,72,60)(13,26,64,61)(14,27,65,62)(15,19,66,63)(16,20,67,55)(17,21,68,56)(18,22,69,57)(28,49,136,37)(29,50,137,38)(30,51,138,39)(31,52,139,40)(32,53,140,41)(33,54,141,42)(34,46,142,43)(35,47,143,44)(36,48,144,45)(73,126,127,107)(74,118,128,108)(75,119,129,100)(76,120,130,101)(77,121,131,102)(78,122,132,103)(79,123,133,104)(80,124,134,105)(81,125,135,106), (1,51)(2,52)(3,53)(4,54)(5,46)(6,47)(7,48)(8,49)(9,50)(10,135)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,118)(26,119)(27,120)(28,96)(29,97)(30,98)(31,99)(32,91)(33,92)(34,93)(35,94)(36,95)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,82)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,100)(62,101)(63,102)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(109,137)(110,138)(111,139)(112,140)(113,141)(114,142)(115,143)(116,144)(117,136), (1,129,85,75)(2,111,86,99)(3,102,87,121)(4,132,88,78)(5,114,89,93)(6,105,90,124)(7,135,82,81)(8,117,83,96)(9,108,84,118)(10,45,70,48)(11,59,71,24)(12,29,72,137)(13,39,64,51)(14,62,65,27)(15,32,66,140)(16,42,67,54)(17,56,68,21)(18,35,69,143)(19,53,63,41)(20,33,55,141)(22,47,57,44)(23,36,58,144)(25,50,60,38)(26,30,61,138)(28,49,136,37)(31,52,139,40)(34,46,142,43)(73,126,127,107)(74,109,128,97)(76,120,130,101)(77,112,131,91)(79,123,133,104)(80,115,134,94)(92,103,113,122)(95,106,116,125)(98,100,110,119), (1,110,85,98)(2,101,86,120)(3,131,87,77)(4,113,88,92)(5,104,89,123)(6,134,90,80)(7,116,82,95)(8,107,83,126)(9,128,84,74)(10,58,70,23)(11,28,71,136)(12,38,72,50)(13,61,64,26)(14,31,65,139)(15,41,66,53)(16,55,67,20)(17,34,68,142)(18,44,69,47)(19,32,63,140)(21,46,56,43)(22,35,57,143)(24,49,59,37)(25,29,60,137)(27,52,62,40)(30,51,138,39)(33,54,141,42)(36,48,144,45)(73,117,127,96)(75,119,129,100)(76,111,130,99)(78,122,132,103)(79,114,133,93)(81,125,135,106)(91,102,112,121)(94,105,115,124)(97,108,109,118), (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) );
G=PermutationGroup([[(1,98,85,110),(2,99,86,111),(3,91,87,112),(4,92,88,113),(5,93,89,114),(6,94,90,115),(7,95,82,116),(8,96,83,117),(9,97,84,109),(10,23,70,58),(11,24,71,59),(12,25,72,60),(13,26,64,61),(14,27,65,62),(15,19,66,63),(16,20,67,55),(17,21,68,56),(18,22,69,57),(28,49,136,37),(29,50,137,38),(30,51,138,39),(31,52,139,40),(32,53,140,41),(33,54,141,42),(34,46,142,43),(35,47,143,44),(36,48,144,45),(73,126,127,107),(74,118,128,108),(75,119,129,100),(76,120,130,101),(77,121,131,102),(78,122,132,103),(79,123,133,104),(80,124,134,105),(81,125,135,106)], [(1,51),(2,52),(3,53),(4,54),(5,46),(6,47),(7,48),(8,49),(9,50),(10,135),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,118),(26,119),(27,120),(28,96),(29,97),(30,98),(31,99),(32,91),(33,92),(34,93),(35,94),(36,95),(37,83),(38,84),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,82),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,100),(62,101),(63,102),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,73),(72,74),(109,137),(110,138),(111,139),(112,140),(113,141),(114,142),(115,143),(116,144),(117,136)], [(1,129,85,75),(2,111,86,99),(3,102,87,121),(4,132,88,78),(5,114,89,93),(6,105,90,124),(7,135,82,81),(8,117,83,96),(9,108,84,118),(10,45,70,48),(11,59,71,24),(12,29,72,137),(13,39,64,51),(14,62,65,27),(15,32,66,140),(16,42,67,54),(17,56,68,21),(18,35,69,143),(19,53,63,41),(20,33,55,141),(22,47,57,44),(23,36,58,144),(25,50,60,38),(26,30,61,138),(28,49,136,37),(31,52,139,40),(34,46,142,43),(73,126,127,107),(74,109,128,97),(76,120,130,101),(77,112,131,91),(79,123,133,104),(80,115,134,94),(92,103,113,122),(95,106,116,125),(98,100,110,119)], [(1,110,85,98),(2,101,86,120),(3,131,87,77),(4,113,88,92),(5,104,89,123),(6,134,90,80),(7,116,82,95),(8,107,83,126),(9,128,84,74),(10,58,70,23),(11,28,71,136),(12,38,72,50),(13,61,64,26),(14,31,65,139),(15,41,66,53),(16,55,67,20),(17,34,68,142),(18,44,69,47),(19,32,63,140),(21,46,56,43),(22,35,57,143),(24,49,59,37),(25,29,60,137),(27,52,62,40),(30,51,138,39),(33,54,141,42),(36,48,144,45),(73,117,127,96),(75,119,129,100),(76,111,130,99),(78,122,132,103),(79,114,133,93),(81,125,135,106),(91,102,112,121),(94,105,115,124),(97,108,109,118)], [(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)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | ··· | 9F | 12A | 12B | 12C | ··· | 12H | 18A | ··· | 18F | 18G | ··· | 18R | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 6 | 1 | 1 | 2 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 4 | ··· | 4 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | A4 | C2×A4 | C2×A4 | C3.A4 | C2×C3.A4 | C2×C3.A4 | D4.A4 | D4.A4 | 2- 1+4⋊C9 |
| kernel | 2- 1+4⋊C9 | C2×Q8⋊C9 | Q8.C18 | C3×2- 1+4 | C6×Q8 | C3×C4○D4 | 2- 1+4 | C2×Q8 | C4○D4 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 6 |
Matrix representation of 2- 1+4⋊C9 ►in GL7(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 14 | 32 | 0 |
| 0 | 0 | 0 | 9 | 14 | 27 | 5 |
| 0 | 0 | 0 | 5 | 0 | 23 | 14 |
| 0 | 0 | 0 | 10 | 32 | 9 | 14 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 9 | 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 0 | 0 | 2 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 36 | 0 | 0 | 0 | 0 | 0 |
| 18 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 26 | 4 | 0 | 0 | 0 | 0 | 0 |
| 7 | 4 | 35 | 0 | 0 | 0 | 0 |
| 33 | 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 10 | 32 |
| 0 | 0 | 0 | 10 | 27 | 10 | 0 |
| 0 | 0 | 0 | 0 | 32 | 27 | 5 |
| 0 | 0 | 0 | 10 | 27 | 27 | 0 |
G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,36,36,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,23,9,5,10,0,0,0,14,14,0,32,0,0,0,32,27,23,9,0,0,0,0,5,14,14],[36,0,9,0,0,0,0,0,36,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,35,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,2,0,0,0,0,0,36,36],[1,24,18,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,36,0,0,0,0,0,0,0,36,0,0],[26,7,33,0,0,0,0,4,4,5,0,0,0,0,0,35,7,0,0,0,0,0,0,0,0,10,0,10,0,0,0,32,27,32,27,0,0,0,10,10,27,27,0,0,0,32,0,5,0] >;
2- 1+4⋊C9 in GAP, Magma, Sage, TeX
2_-^{1+4}\rtimes C_9 % in TeX
G:=Group("ES-(2,2):C9"); // GroupNames label
G:=SmallGroup(288,349);
// by ID
G=gap.SmallGroup(288,349);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,79,648,172,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=e^9=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations