metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D18, (C2×D4).5D9, (C2×C18).7D4, (C6×D4).19S3, (C2×C4).15D18, C18.48(C2×D4), Dic9⋊C4⋊14C2, (D4×C18).10C2, (C2×C12).215D6, (C22×C6).48D6, C18.29(C4○D4), C18.D4⋊8C2, (C2×C18).50C23, (C2×C36).60C22, (C22×Dic9)⋊5C2, C22.4(C9⋊D4), C9⋊5(C22.D4), C6.86(D4⋊2S3), C2.15(D4⋊2D9), C3.(C23.23D6), C22.57(C22×D9), (C22×C18).18C22, (C2×Dic9).15C22, C2.11(C2×C9⋊D4), C6.95(C2×C3⋊D4), (C2×C6).4(C3⋊D4), (C2×C6).207(C22×S3), SmallGroup(288,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D18
G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 404 in 117 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C18, C18, C18, C2×Dic3, C2×C12, C3×D4, C22×C6, C22.D4, Dic9, C36, C2×C18, C2×C18, C2×C18, Dic3⋊C4, C6.D4, C22×Dic3, C6×D4, C2×Dic9, C2×Dic9, C2×C36, D4×C9, C22×C18, C23.23D6, Dic9⋊C4, C18.D4, C18.D4, C22×Dic9, D4×C18, C23.23D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C22.D4, D18, D4⋊2S3, C2×C3⋊D4, C9⋊D4, C22×D9, C23.23D6, D4⋊2D9, C2×C9⋊D4, C23.23D18
►On 144 points
Generators in S144
(1 132)(2 142)(3 134)(4 144)(5 136)(6 128)(7 138)(8 130)(9 140)(10 143)(11 135)(12 127)(13 137)(14 129)(15 139)(16 131)(17 141)(18 133)(19 64)(20 56)(21 66)(22 58)(23 68)(24 60)(25 70)(26 62)(27 72)(28 65)(29 57)(30 67)(31 59)(32 69)(33 61)(34 71)(35 63)(36 55)(37 112)(38 81)(39 114)(40 83)(41 116)(42 85)(43 118)(44 87)(45 120)(46 89)(47 122)(48 73)(49 124)(50 75)(51 126)(52 77)(53 110)(54 79)(74 99)(76 101)(78 103)(80 105)(82 107)(84 91)(86 93)(88 95)(90 97)(92 117)(94 119)(96 121)(98 123)(100 125)(102 109)(104 111)(106 113)(108 115)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)(37 96)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 91)(51 92)(52 93)(53 94)(54 95)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 127)(70 128)(71 129)(72 130)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(19 36)(20 28)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)(37 105)(38 106)(39 107)(40 108)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(73 123)(74 124)(75 125)(76 126)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(127 136)(128 137)(129 138)(130 139)(131 140)(132 141)(133 142)(134 143)(135 144)
(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)
(1 112 28 89)(2 120 29 79)(3 110 30 87)(4 118 31 77)(5 126 32 85)(6 116 33 75)(7 124 34 83)(8 114 35 73)(9 122 36 81)(10 78 22 119)(11 86 23 109)(12 76 24 117)(13 84 25 125)(14 74 26 115)(15 82 27 123)(16 90 19 113)(17 80 20 121)(18 88 21 111)(37 56 46 141)(38 131 47 64)(39 72 48 139)(40 129 49 62)(41 70 50 137)(42 127 51 60)(43 68 52 135)(44 143 53 58)(45 66 54 133)(55 106 140 97)(57 104 142 95)(59 102 144 93)(61 100 128 91)(63 98 130 107)(65 96 132 105)(67 94 134 103)(69 92 136 101)(71 108 138 99)
G:=sub<Sym(144)| (1,132)(2,142)(3,134)(4,144)(5,136)(6,128)(7,138)(8,130)(9,140)(10,143)(11,135)(12,127)(13,137)(14,129)(15,139)(16,131)(17,141)(18,133)(19,64)(20,56)(21,66)(22,58)(23,68)(24,60)(25,70)(26,62)(27,72)(28,65)(29,57)(30,67)(31,59)(32,69)(33,61)(34,71)(35,63)(36,55)(37,112)(38,81)(39,114)(40,83)(41,116)(42,85)(43,118)(44,87)(45,120)(46,89)(47,122)(48,73)(49,124)(50,75)(51,126)(52,77)(53,110)(54,79)(74,99)(76,101)(78,103)(80,105)(82,107)(84,91)(86,93)(88,95)(90,97)(92,117)(94,119)(96,121)(98,123)(100,125)(102,109)(104,111)(106,113)(108,115), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,91)(51,92)(52,93)(53,94)(54,95)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,127)(70,128)(71,129)(72,130)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(19,36)(20,28)(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(37,105)(38,106)(39,107)(40,108)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(73,123)(74,124)(75,125)(76,126)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141)(133,142)(134,143)(135,144), (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), (1,112,28,89)(2,120,29,79)(3,110,30,87)(4,118,31,77)(5,126,32,85)(6,116,33,75)(7,124,34,83)(8,114,35,73)(9,122,36,81)(10,78,22,119)(11,86,23,109)(12,76,24,117)(13,84,25,125)(14,74,26,115)(15,82,27,123)(16,90,19,113)(17,80,20,121)(18,88,21,111)(37,56,46,141)(38,131,47,64)(39,72,48,139)(40,129,49,62)(41,70,50,137)(42,127,51,60)(43,68,52,135)(44,143,53,58)(45,66,54,133)(55,106,140,97)(57,104,142,95)(59,102,144,93)(61,100,128,91)(63,98,130,107)(65,96,132,105)(67,94,134,103)(69,92,136,101)(71,108,138,99)>;
G:=Group( (1,132)(2,142)(3,134)(4,144)(5,136)(6,128)(7,138)(8,130)(9,140)(10,143)(11,135)(12,127)(13,137)(14,129)(15,139)(16,131)(17,141)(18,133)(19,64)(20,56)(21,66)(22,58)(23,68)(24,60)(25,70)(26,62)(27,72)(28,65)(29,57)(30,67)(31,59)(32,69)(33,61)(34,71)(35,63)(36,55)(37,112)(38,81)(39,114)(40,83)(41,116)(42,85)(43,118)(44,87)(45,120)(46,89)(47,122)(48,73)(49,124)(50,75)(51,126)(52,77)(53,110)(54,79)(74,99)(76,101)(78,103)(80,105)(82,107)(84,91)(86,93)(88,95)(90,97)(92,117)(94,119)(96,121)(98,123)(100,125)(102,109)(104,111)(106,113)(108,115), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,91)(51,92)(52,93)(53,94)(54,95)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,127)(70,128)(71,129)(72,130)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(19,36)(20,28)(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(37,105)(38,106)(39,107)(40,108)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(73,123)(74,124)(75,125)(76,126)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141)(133,142)(134,143)(135,144), (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), (1,112,28,89)(2,120,29,79)(3,110,30,87)(4,118,31,77)(5,126,32,85)(6,116,33,75)(7,124,34,83)(8,114,35,73)(9,122,36,81)(10,78,22,119)(11,86,23,109)(12,76,24,117)(13,84,25,125)(14,74,26,115)(15,82,27,123)(16,90,19,113)(17,80,20,121)(18,88,21,111)(37,56,46,141)(38,131,47,64)(39,72,48,139)(40,129,49,62)(41,70,50,137)(42,127,51,60)(43,68,52,135)(44,143,53,58)(45,66,54,133)(55,106,140,97)(57,104,142,95)(59,102,144,93)(61,100,128,91)(63,98,130,107)(65,96,132,105)(67,94,134,103)(69,92,136,101)(71,108,138,99) );
G=PermutationGroup([[(1,132),(2,142),(3,134),(4,144),(5,136),(6,128),(7,138),(8,130),(9,140),(10,143),(11,135),(12,127),(13,137),(14,129),(15,139),(16,131),(17,141),(18,133),(19,64),(20,56),(21,66),(22,58),(23,68),(24,60),(25,70),(26,62),(27,72),(28,65),(29,57),(30,67),(31,59),(32,69),(33,61),(34,71),(35,63),(36,55),(37,112),(38,81),(39,114),(40,83),(41,116),(42,85),(43,118),(44,87),(45,120),(46,89),(47,122),(48,73),(49,124),(50,75),(51,126),(52,77),(53,110),(54,79),(74,99),(76,101),(78,103),(80,105),(82,107),(84,91),(86,93),(88,95),(90,97),(92,117),(94,119),(96,121),(98,123),(100,125),(102,109),(104,111),(106,113),(108,115)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,19),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29),(37,96),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,91),(51,92),(52,93),(53,94),(54,95),(55,131),(56,132),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,140),(65,141),(66,142),(67,143),(68,144),(69,127),(70,128),(71,129),(72,130),(73,82),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(109,118),(110,119),(111,120),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126)], [(1,17),(2,18),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(19,36),(20,28),(21,29),(22,30),(23,31),(24,32),(25,33),(26,34),(27,35),(37,105),(38,106),(39,107),(40,108),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(73,123),(74,124),(75,125),(76,126),(77,109),(78,110),(79,111),(80,112),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(127,136),(128,137),(129,138),(130,139),(131,140),(132,141),(133,142),(134,143),(135,144)], [(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)], [(1,112,28,89),(2,120,29,79),(3,110,30,87),(4,118,31,77),(5,126,32,85),(6,116,33,75),(7,124,34,83),(8,114,35,73),(9,122,36,81),(10,78,22,119),(11,86,23,109),(12,76,24,117),(13,84,25,125),(14,74,26,115),(15,82,27,123),(16,90,19,113),(17,80,20,121),(18,88,21,111),(37,56,46,141),(38,131,47,64),(39,72,48,139),(40,129,49,62),(41,70,50,137),(42,127,51,60),(43,68,52,135),(44,143,53,58),(45,66,54,133),(55,106,140,97),(57,104,142,95),(59,102,144,93),(61,100,128,91),(63,98,130,107),(65,96,132,105),(67,94,134,103),(69,92,136,101),(71,108,138,99)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 3 | 4 | 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 | 2 | 2 | 4 | 2 | 4 | 18 | 18 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | D4⋊2S3 | D4⋊2D9 |
| kernel | C23.23D18 | Dic9⋊C4 | C18.D4 | C22×Dic9 | D4×C18 | C6×D4 | C2×C18 | C2×C12 | C22×C6 | C18 | C2×D4 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 3 | 4 | 3 | 6 | 12 | 2 | 6 |
Matrix representation of C23.23D18 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 9 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 6 | 0 | 0 |
| 0 | 0 | 31 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 24 | 36 |
| 36 | 29 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 27 | 0 | 0 |
| 0 | 0 | 32 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 4 | 6 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,9,0,0,0,0,0,36,0,0,0,0,0,0,26,31,0,0,0,0,6,20,0,0,0,0,0,0,1,24,0,0,0,0,0,36],[36,28,0,0,0,0,29,1,0,0,0,0,0,0,32,32,0,0,0,0,27,5,0,0,0,0,0,0,31,4,0,0,0,0,0,6] >;
C23.23D18 in GAP, Magma, Sage, TeX
C_2^3._{23}D_{18} % in TeX
G:=Group("C2^3.23D18"); // GroupNames label
G:=SmallGroup(288,145);
// by ID
G=gap.SmallGroup(288,145);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^18=1,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations