direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×Dic9, C23.3D9, C18.9C23, C22.11D18, C18⋊2(C2×C4), (C2×C18)⋊3C4, C9⋊2(C22×C4), (C2×C6).28D6, (C22×C6).7S3, (C2×C6).6Dic3, C2.2(C22×D9), C3.(C22×Dic3), C6.27(C22×S3), (C22×C18).3C2, C6.11(C2×Dic3), (C2×C18).12C22, SmallGroup(144,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C22×Dic9 |
Generators and relations for C22×Dic9
G = < a,b,c,d | a2=b2=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 191 in 81 conjugacy classes, 59 normal (10 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C6, C6 [×6], C2×C4 [×6], C23, C9, Dic3 [×4], C2×C6 [×7], C22×C4, C18, C18 [×6], C2×Dic3 [×6], C22×C6, Dic9 [×4], C2×C18 [×7], C22×Dic3, C2×Dic9 [×6], C22×C18, C22×Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, D9, C2×Dic3 [×6], C22×S3, Dic9 [×4], D18 [×3], C22×Dic3, C2×Dic9 [×6], C22×D9, C22×Dic9
►Regular action on 144 points
Generators in S144
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 37)(16 38)(17 39)(18 40)(19 70)(20 71)(21 72)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(73 124)(74 125)(75 126)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 19)(15 20)(16 21)(17 22)(18 23)(37 71)(38 72)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(73 105)(74 106)(75 107)(76 108)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 127)(120 128)(121 129)(122 130)(123 131)(124 132)(125 133)(126 134)
(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 93 10 102)(2 92 11 101)(3 91 12 100)(4 108 13 99)(5 107 14 98)(6 106 15 97)(7 105 16 96)(8 104 17 95)(9 103 18 94)(19 84 28 75)(20 83 29 74)(21 82 30 73)(22 81 31 90)(23 80 32 89)(24 79 33 88)(25 78 34 87)(26 77 35 86)(27 76 36 85)(37 142 46 133)(38 141 47 132)(39 140 48 131)(40 139 49 130)(41 138 50 129)(42 137 51 128)(43 136 52 127)(44 135 53 144)(45 134 54 143)(55 114 64 123)(56 113 65 122)(57 112 66 121)(58 111 67 120)(59 110 68 119)(60 109 69 118)(61 126 70 117)(62 125 71 116)(63 124 72 115)
G:=sub<Sym(144)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,37)(16,38)(17,39)(18,40)(19,70)(20,71)(21,72)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(73,124)(74,125)(75,126)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,136)(92,137)(93,138)(94,139)(95,140)(96,141)(97,142)(98,143)(99,144)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,19)(15,20)(16,21)(17,22)(18,23)(37,71)(38,72)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(73,105)(74,106)(75,107)(76,108)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,127)(120,128)(121,129)(122,130)(123,131)(124,132)(125,133)(126,134), (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), (1,93,10,102)(2,92,11,101)(3,91,12,100)(4,108,13,99)(5,107,14,98)(6,106,15,97)(7,105,16,96)(8,104,17,95)(9,103,18,94)(19,84,28,75)(20,83,29,74)(21,82,30,73)(22,81,31,90)(23,80,32,89)(24,79,33,88)(25,78,34,87)(26,77,35,86)(27,76,36,85)(37,142,46,133)(38,141,47,132)(39,140,48,131)(40,139,49,130)(41,138,50,129)(42,137,51,128)(43,136,52,127)(44,135,53,144)(45,134,54,143)(55,114,64,123)(56,113,65,122)(57,112,66,121)(58,111,67,120)(59,110,68,119)(60,109,69,118)(61,126,70,117)(62,125,71,116)(63,124,72,115)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,37)(16,38)(17,39)(18,40)(19,70)(20,71)(21,72)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(73,124)(74,125)(75,126)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,136)(92,137)(93,138)(94,139)(95,140)(96,141)(97,142)(98,143)(99,144)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,19)(15,20)(16,21)(17,22)(18,23)(37,71)(38,72)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(73,105)(74,106)(75,107)(76,108)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,127)(120,128)(121,129)(122,130)(123,131)(124,132)(125,133)(126,134), (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), (1,93,10,102)(2,92,11,101)(3,91,12,100)(4,108,13,99)(5,107,14,98)(6,106,15,97)(7,105,16,96)(8,104,17,95)(9,103,18,94)(19,84,28,75)(20,83,29,74)(21,82,30,73)(22,81,31,90)(23,80,32,89)(24,79,33,88)(25,78,34,87)(26,77,35,86)(27,76,36,85)(37,142,46,133)(38,141,47,132)(39,140,48,131)(40,139,49,130)(41,138,50,129)(42,137,51,128)(43,136,52,127)(44,135,53,144)(45,134,54,143)(55,114,64,123)(56,113,65,122)(57,112,66,121)(58,111,67,120)(59,110,68,119)(60,109,69,118)(61,126,70,117)(62,125,71,116)(63,124,72,115) );
G=PermutationGroup([(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,37),(16,38),(17,39),(18,40),(19,70),(20,71),(21,72),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(73,124),(74,125),(75,126),(76,109),(77,110),(78,111),(79,112),(80,113),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,121),(89,122),(90,123),(91,136),(92,137),(93,138),(94,139),(95,140),(96,141),(97,142),(98,143),(99,144),(100,127),(101,128),(102,129),(103,130),(104,131),(105,132),(106,133),(107,134),(108,135)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,19),(15,20),(16,21),(17,22),(18,23),(37,71),(38,72),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(73,105),(74,106),(75,107),(76,108),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(118,144),(119,127),(120,128),(121,129),(122,130),(123,131),(124,132),(125,133),(126,134)], [(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)], [(1,93,10,102),(2,92,11,101),(3,91,12,100),(4,108,13,99),(5,107,14,98),(6,106,15,97),(7,105,16,96),(8,104,17,95),(9,103,18,94),(19,84,28,75),(20,83,29,74),(21,82,30,73),(22,81,31,90),(23,80,32,89),(24,79,33,88),(25,78,34,87),(26,77,35,86),(27,76,36,85),(37,142,46,133),(38,141,47,132),(39,140,48,131),(40,139,49,130),(41,138,50,129),(42,137,51,128),(43,136,52,127),(44,135,53,144),(45,134,54,143),(55,114,64,123),(56,113,65,122),(57,112,66,121),(58,111,67,120),(59,110,68,119),(60,109,69,118),(61,126,70,117),(62,125,71,116),(63,124,72,115)])
C22×Dic9 is a maximal subgroup of
C18.C42 C23.16D18 C22⋊2Dic18 Dic9⋊4D4 C22.4D36 C23.23D18 Dic9⋊D4 C22×C4×D9 Dic9⋊A4
C22×Dic9 is a maximal quotient of
C23.26D18 D4.Dic9
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 6A | ··· | 6G | 9A | 9B | 9C | 18A | ··· | 18U |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 18 | ··· | 18 |
| size | 1 | 1 | ··· | 1 | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | D9 | Dic9 | D18 |
| kernel | C22×Dic9 | C2×Dic9 | C22×C18 | C2×C18 | C22×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 8 | 1 | 4 | 3 | 3 | 12 | 9 |
Matrix representation of C22×Dic9 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 6 | 17 |
| 0 | 0 | 20 | 26 |
| 36 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 7 | 30 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,36,0,0,0,0,6,20,0,0,17,26],[36,0,0,0,0,31,0,0,0,0,7,7,0,0,14,30] >;
C22×Dic9 in GAP, Magma, Sage, TeX
C_2^2\times {\rm Dic}_9 % in TeX
G:=Group("C2^2xDic9"); // GroupNames label
G:=SmallGroup(144,45);
// by ID
G=gap.SmallGroup(144,45);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations