metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.28D18, D18⋊C4⋊2C2, (C22×C4)⋊5D9, Dic9⋊C4⋊3C2, (C22×C36)⋊2C2, (C2×C4).69D18, C18.43(C2×D4), (C2×C18).37D4, (C2×C12).345D6, C18.18(C4○D4), C6.88(C4○D12), C18.D4⋊6C2, (C2×C18).47C23, (C22×C12).12S3, (C2×C36).77C22, (C22×C6).141D6, C9⋊4(C22.D4), C22.9(C9⋊D4), C3.(C23.28D6), C2.18(D36⋊5C2), (C22×D9).9C22, C22.55(C22×D9), (C22×C18).39C22, (C2×Dic9).13C22, C2.6(C2×C9⋊D4), (C2×C9⋊D4).6C2, C6.90(C2×C3⋊D4), (C2×C6).76(C3⋊D4), (C2×C6).204(C22×S3), SmallGroup(288,139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.28D18
G = < a,b,c,d,e | a2=b2=c2=1, d18=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd17 >
Subgroups: 484 in 117 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22.D4, Dic9, C36, D18, C2×C18, C2×C18, C2×C18, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, C2×C36, C22×D9, C22×C18, C23.28D6, Dic9⋊C4, D18⋊C4, C18.D4, C2×C9⋊D4, C22×C36, C23.28D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C22.D4, D18, C4○D12, C2×C3⋊D4, C9⋊D4, C22×D9, C23.28D6, D36⋊5C2, C2×C9⋊D4, C23.28D18
►On 144 points
Generators in S144
(1 117)(2 118)(3 119)(4 120)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(28 144)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 73)(36 74)(37 142)(38 143)(39 144)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(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 18 93 74)(2 73 94 17)(3 16 95 108)(4 107 96 15)(5 14 97 106)(6 105 98 13)(7 12 99 104)(8 103 100 11)(9 10 101 102)(19 36 75 92)(20 91 76 35)(21 34 77 90)(22 89 78 33)(23 32 79 88)(24 87 80 31)(25 30 81 86)(26 85 82 29)(27 28 83 84)(37 109 124 58)(38 57 125 144)(39 143 126 56)(40 55 127 142)(41 141 128 54)(42 53 129 140)(43 139 130 52)(44 51 131 138)(45 137 132 50)(46 49 133 136)(47 135 134 48)(59 123 110 72)(60 71 111 122)(61 121 112 70)(62 69 113 120)(63 119 114 68)(64 67 115 118)(65 117 116 66)
G:=sub<Sym(144)| (1,117)(2,118)(3,119)(4,120)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,144)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,73)(36,74)(37,142)(38,143)(39,144)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (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), (1,18,93,74)(2,73,94,17)(3,16,95,108)(4,107,96,15)(5,14,97,106)(6,105,98,13)(7,12,99,104)(8,103,100,11)(9,10,101,102)(19,36,75,92)(20,91,76,35)(21,34,77,90)(22,89,78,33)(23,32,79,88)(24,87,80,31)(25,30,81,86)(26,85,82,29)(27,28,83,84)(37,109,124,58)(38,57,125,144)(39,143,126,56)(40,55,127,142)(41,141,128,54)(42,53,129,140)(43,139,130,52)(44,51,131,138)(45,137,132,50)(46,49,133,136)(47,135,134,48)(59,123,110,72)(60,71,111,122)(61,121,112,70)(62,69,113,120)(63,119,114,68)(64,67,115,118)(65,117,116,66)>;
G:=Group( (1,117)(2,118)(3,119)(4,120)(5,121)(6,122)(7,123)(8,124)(9,125)(10,126)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,137)(22,138)(23,139)(24,140)(25,141)(26,142)(27,143)(28,144)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,73)(36,74)(37,142)(38,143)(39,144)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (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), (1,18,93,74)(2,73,94,17)(3,16,95,108)(4,107,96,15)(5,14,97,106)(6,105,98,13)(7,12,99,104)(8,103,100,11)(9,10,101,102)(19,36,75,92)(20,91,76,35)(21,34,77,90)(22,89,78,33)(23,32,79,88)(24,87,80,31)(25,30,81,86)(26,85,82,29)(27,28,83,84)(37,109,124,58)(38,57,125,144)(39,143,126,56)(40,55,127,142)(41,141,128,54)(42,53,129,140)(43,139,130,52)(44,51,131,138)(45,137,132,50)(46,49,133,136)(47,135,134,48)(59,123,110,72)(60,71,111,122)(61,121,112,70)(62,69,113,120)(63,119,114,68)(64,67,115,118)(65,117,116,66) );
G=PermutationGroup([[(1,117),(2,118),(3,119),(4,120),(5,121),(6,122),(7,123),(8,124),(9,125),(10,126),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,137),(22,138),(23,139),(24,140),(25,141),(26,142),(27,143),(28,144),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,73),(36,74),(37,142),(38,143),(39,144),(40,109),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,121),(53,122),(54,123),(55,124),(56,125),(57,126),(58,127),(59,128),(60,129),(61,130),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144)], [(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)], [(1,18,93,74),(2,73,94,17),(3,16,95,108),(4,107,96,15),(5,14,97,106),(6,105,98,13),(7,12,99,104),(8,103,100,11),(9,10,101,102),(19,36,75,92),(20,91,76,35),(21,34,77,90),(22,89,78,33),(23,32,79,88),(24,87,80,31),(25,30,81,86),(26,85,82,29),(27,28,83,84),(37,109,124,58),(38,57,125,144),(39,143,126,56),(40,55,127,142),(41,141,128,54),(42,53,129,140),(43,139,130,52),(44,51,131,138),(45,137,132,50),(46,49,133,136),(47,135,134,48),(59,123,110,72),(60,71,111,122),(61,121,112,70),(62,69,113,120),(63,119,114,68),(64,67,115,118),(65,117,116,66)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | D18 | C4○D12 | C9⋊D4 | D36⋊5C2 |
| kernel | C23.28D18 | Dic9⋊C4 | D18⋊C4 | C18.D4 | C2×C9⋊D4 | C22×C36 | C22×C12 | C2×C18 | C2×C12 | C22×C6 | C18 | C22×C4 | C2×C6 | C2×C4 | C23 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 3 | 4 | 6 | 3 | 8 | 12 | 24 |
Matrix representation of C23.28D18 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 30 | 14 |
| 0 | 0 | 23 | 7 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 30 | 30 | 0 | 0 |
| 7 | 23 | 0 | 0 |
| 0 | 0 | 29 | 33 |
| 0 | 0 | 4 | 25 |
| 30 | 14 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 25 | 4 |
| 0 | 0 | 29 | 12 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,30,23,0,0,14,7],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[30,7,0,0,30,23,0,0,0,0,29,4,0,0,33,25],[30,7,0,0,14,7,0,0,0,0,25,29,0,0,4,12] >;
C23.28D18 in GAP, Magma, Sage, TeX
C_2^3._{28}D_{18} % in TeX
G:=Group("C2^3.28D18"); // GroupNames label
G:=SmallGroup(288,139);
// by ID
G=gap.SmallGroup(288,139);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^18=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^17>;
// generators/relations