direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C22.D4, C4⋊C4⋊4C18, C2.7(D4×C18), C6.70(C6×D4), C22⋊C4⋊4C18, (C22×C36)⋊5C2, (C22×C4)⋊5C18, (C6×D4).11C6, (C2×D4).4C18, C18.70(C2×D4), (C2×C18).24D4, C22.4(D4×C9), (D4×C18).11C2, C23.3(C2×C18), C18.43(C4○D4), (C2×C36).79C22, (C22×C12).14C6, (C2×C18).78C23, (C22×C18).29C22, C22.13(C22×C18), (C9×C4⋊C4)⋊13C2, C2.6(C9×C4○D4), (C3×C4⋊C4).14C6, C6.43(C3×C4○D4), (C2×C6).28(C3×D4), (C9×C22⋊C4)⋊12C2, (C2×C4).10(C2×C18), (C2×C12).82(C2×C6), (C3×C22⋊C4).8C6, C3.(C3×C22.D4), (C2×C6).83(C22×C6), (C22×C6).48(C2×C6), (C3×C22.D4).C3, SmallGroup(288,173)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C22.D4
G = < a,b,c,d,e | a9=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 174 in 117 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C18, C18, C18, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22.D4, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C2×C36, C2×C36, C2×C36, D4×C9, C22×C18, C3×C22.D4, C9×C22⋊C4, C9×C22⋊C4, C9×C4⋊C4, C22×C36, D4×C18, C9×C22.D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C4○D4, C18, C3×D4, C22×C6, C22.D4, C2×C18, C6×D4, C3×C4○D4, D4×C9, C22×C18, C3×C22.D4, D4×C18, C9×C4○D4, C9×C22.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)
(1 106)(2 107)(3 108)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 68)(11 69)(12 70)(13 71)(14 72)(15 64)(16 65)(17 66)(18 67)(19 59)(20 60)(21 61)(22 62)(23 63)(24 55)(25 56)(26 57)(27 58)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 109)(36 110)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 118)(44 119)(45 120)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 91)(53 92)(54 93)(73 143)(74 144)(75 136)(76 137)(77 138)(78 139)(79 140)(80 141)(81 142)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 127)(89 128)(90 129)
(1 42)(2 43)(3 44)(4 45)(5 37)(6 38)(7 39)(8 40)(9 41)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 144)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 46)(36 47)(55 79)(56 80)(57 81)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 82)(72 83)(91 115)(92 116)(93 117)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 118)(108 119)
(1 57 54 70)(2 58 46 71)(3 59 47 72)(4 60 48 64)(5 61 49 65)(6 62 50 66)(7 63 51 67)(8 55 52 68)(9 56 53 69)(10 124 24 115)(11 125 25 116)(12 126 26 117)(13 118 27 109)(14 119 19 110)(15 120 20 111)(16 121 21 112)(17 122 22 113)(18 123 23 114)(28 84 45 75)(29 85 37 76)(30 86 38 77)(31 87 39 78)(32 88 40 79)(33 89 41 80)(34 90 42 81)(35 82 43 73)(36 83 44 74)(91 127 104 140)(92 128 105 141)(93 129 106 142)(94 130 107 143)(95 131 108 144)(96 132 100 136)(97 133 101 137)(98 134 102 138)(99 135 103 139)
(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(55 88)(56 89)(57 90)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 73)(72 74)(91 115)(92 116)(93 117)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 118)(108 119)(127 140)(128 141)(129 142)(130 143)(131 144)(132 136)(133 137)(134 138)(135 139)
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), (1,106)(2,107)(3,108)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,68)(11,69)(12,70)(13,71)(14,72)(15,64)(16,65)(17,66)(18,67)(19,59)(20,60)(21,61)(22,62)(23,63)(24,55)(25,56)(26,57)(27,58)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,109)(36,110)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,118)(44,119)(45,120)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,91)(53,92)(54,93)(73,143)(74,144)(75,136)(76,137)(77,138)(78,139)(79,140)(80,141)(81,142)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,127)(89,128)(90,129), (1,42)(2,43)(3,44)(4,45)(5,37)(6,38)(7,39)(8,40)(9,41)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,144)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47)(55,79)(56,80)(57,81)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,82)(72,83)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119), (1,57,54,70)(2,58,46,71)(3,59,47,72)(4,60,48,64)(5,61,49,65)(6,62,50,66)(7,63,51,67)(8,55,52,68)(9,56,53,69)(10,124,24,115)(11,125,25,116)(12,126,26,117)(13,118,27,109)(14,119,19,110)(15,120,20,111)(16,121,21,112)(17,122,22,113)(18,123,23,114)(28,84,45,75)(29,85,37,76)(30,86,38,77)(31,87,39,78)(32,88,40,79)(33,89,41,80)(34,90,42,81)(35,82,43,73)(36,83,44,74)(91,127,104,140)(92,128,105,141)(93,129,106,142)(94,130,107,143)(95,131,108,144)(96,132,100,136)(97,133,101,137)(98,134,102,138)(99,135,103,139), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(55,88)(56,89)(57,90)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139)>;
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), (1,106)(2,107)(3,108)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,68)(11,69)(12,70)(13,71)(14,72)(15,64)(16,65)(17,66)(18,67)(19,59)(20,60)(21,61)(22,62)(23,63)(24,55)(25,56)(26,57)(27,58)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,109)(36,110)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,118)(44,119)(45,120)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,91)(53,92)(54,93)(73,143)(74,144)(75,136)(76,137)(77,138)(78,139)(79,140)(80,141)(81,142)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,127)(89,128)(90,129), (1,42)(2,43)(3,44)(4,45)(5,37)(6,38)(7,39)(8,40)(9,41)(10,127)(11,128)(12,129)(13,130)(14,131)(15,132)(16,133)(17,134)(18,135)(19,144)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47)(55,79)(56,80)(57,81)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,82)(72,83)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119), (1,57,54,70)(2,58,46,71)(3,59,47,72)(4,60,48,64)(5,61,49,65)(6,62,50,66)(7,63,51,67)(8,55,52,68)(9,56,53,69)(10,124,24,115)(11,125,25,116)(12,126,26,117)(13,118,27,109)(14,119,19,110)(15,120,20,111)(16,121,21,112)(17,122,22,113)(18,123,23,114)(28,84,45,75)(29,85,37,76)(30,86,38,77)(31,87,39,78)(32,88,40,79)(33,89,41,80)(34,90,42,81)(35,82,43,73)(36,83,44,74)(91,127,104,140)(92,128,105,141)(93,129,106,142)(94,130,107,143)(95,131,108,144)(96,132,100,136)(97,133,101,137)(98,134,102,138)(99,135,103,139), (10,24)(11,25)(12,26)(13,27)(14,19)(15,20)(16,21)(17,22)(18,23)(55,88)(56,89)(57,90)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,73)(72,74)(91,115)(92,116)(93,117)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,118)(108,119)(127,140)(128,141)(129,142)(130,143)(131,144)(132,136)(133,137)(134,138)(135,139) );
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)], [(1,106),(2,107),(3,108),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,68),(11,69),(12,70),(13,71),(14,72),(15,64),(16,65),(17,66),(18,67),(19,59),(20,60),(21,61),(22,62),(23,63),(24,55),(25,56),(26,57),(27,58),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,109),(36,110),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,118),(44,119),(45,120),(46,94),(47,95),(48,96),(49,97),(50,98),(51,99),(52,91),(53,92),(54,93),(73,143),(74,144),(75,136),(76,137),(77,138),(78,139),(79,140),(80,141),(81,142),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,127),(89,128),(90,129)], [(1,42),(2,43),(3,44),(4,45),(5,37),(6,38),(7,39),(8,40),(9,41),(10,127),(11,128),(12,129),(13,130),(14,131),(15,132),(16,133),(17,134),(18,135),(19,144),(20,136),(21,137),(22,138),(23,139),(24,140),(25,141),(26,142),(27,143),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,46),(36,47),(55,79),(56,80),(57,81),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,82),(72,83),(91,115),(92,116),(93,117),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,118),(108,119)], [(1,57,54,70),(2,58,46,71),(3,59,47,72),(4,60,48,64),(5,61,49,65),(6,62,50,66),(7,63,51,67),(8,55,52,68),(9,56,53,69),(10,124,24,115),(11,125,25,116),(12,126,26,117),(13,118,27,109),(14,119,19,110),(15,120,20,111),(16,121,21,112),(17,122,22,113),(18,123,23,114),(28,84,45,75),(29,85,37,76),(30,86,38,77),(31,87,39,78),(32,88,40,79),(33,89,41,80),(34,90,42,81),(35,82,43,73),(36,83,44,74),(91,127,104,140),(92,128,105,141),(93,129,106,142),(94,130,107,143),(95,131,108,144),(96,132,100,136),(97,133,101,137),(98,134,102,138),(99,135,103,139)], [(10,24),(11,25),(12,26),(13,27),(14,19),(15,20),(16,21),(17,22),(18,23),(55,88),(56,89),(57,90),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,73),(72,74),(91,115),(92,116),(93,117),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,118),(108,119),(127,140),(128,141),(129,142),(130,143),(131,144),(132,136),(133,137),(134,138),(135,139)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12N | 18A | ··· | 18R | 18S | ··· | 18AD | 18AE | ··· | 18AJ | 36A | ··· | 36X | 36Y | ··· | 36AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | D4×C9 | C9×C4○D4 |
| kernel | C9×C22.D4 | C9×C22⋊C4 | C9×C4⋊C4 | C22×C36 | D4×C18 | C3×C22.D4 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C18 | C18 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 2 | 6 | 4 | 2 | 2 | 6 | 18 | 12 | 6 | 6 | 2 | 4 | 4 | 8 | 12 | 24 |
Matrix representation of C9×C22.D4 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 6 | 22 |
| 0 | 0 | 27 | 31 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 31 | 0 | 0 | 0 |
| 12 | 6 | 0 | 0 |
| 0 | 0 | 36 | 21 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 23 | 36 |
G:=sub<GL(4,GF(37))| [26,0,0,0,0,26,0,0,0,0,9,0,0,0,0,9],[36,0,0,0,0,36,0,0,0,0,6,27,0,0,22,31],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[31,12,0,0,0,6,0,0,0,0,36,0,0,0,21,1],[1,0,0,0,1,36,0,0,0,0,1,23,0,0,0,36] >;
C9×C22.D4 in GAP, Magma, Sage, TeX
C_9\times C_2^2.D_4
% in TeX
G:=Group("C9xC2^2.D4"); // GroupNames label
G:=SmallGroup(288,173);
// by ID
G=gap.SmallGroup(288,173);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,1094,142,360]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations