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, C3, C4, C22, C6, C6, C2×C4, C23, C9, Dic3, C2×C6, C22×C4, C18, C18, C2×Dic3, C22×C6, Dic9, C2×C18, C22×Dic3, C2×Dic9, C22×C18, C22×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C2×Dic3, C22×S3, Dic9, D18, C22×Dic3, C2×Dic9, C22×D9, C22×Dic9
►Regular action on 144 points
Generators in S144
(1 54)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 55)(34 56)(35 57)(36 58)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)
(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 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(73 104)(74 105)(75 106)(76 107)(77 108)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 127)(117 128)(118 129)(119 130)(120 131)(121 132)(122 133)(123 134)(124 135)(125 136)(126 137)
(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 94 10 103)(2 93 11 102)(3 92 12 101)(4 91 13 100)(5 108 14 99)(6 107 15 98)(7 106 16 97)(8 105 17 96)(9 104 18 95)(19 86 28 77)(20 85 29 76)(21 84 30 75)(22 83 31 74)(23 82 32 73)(24 81 33 90)(25 80 34 89)(26 79 35 88)(27 78 36 87)(37 139 46 130)(38 138 47 129)(39 137 48 128)(40 136 49 127)(41 135 50 144)(42 134 51 143)(43 133 52 142)(44 132 53 141)(45 131 54 140)(55 120 64 111)(56 119 65 110)(57 118 66 109)(58 117 67 126)(59 116 68 125)(60 115 69 124)(61 114 70 123)(62 113 71 122)(63 112 72 121)
G:=sub<Sym(144)| (1,54)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,55)(34,56)(35,57)(36,58)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136), (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)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(73,104)(74,105)(75,106)(76,107)(77,108)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137), (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), (1,94,10,103)(2,93,11,102)(3,92,12,101)(4,91,13,100)(5,108,14,99)(6,107,15,98)(7,106,16,97)(8,105,17,96)(9,104,18,95)(19,86,28,77)(20,85,29,76)(21,84,30,75)(22,83,31,74)(23,82,32,73)(24,81,33,90)(25,80,34,89)(26,79,35,88)(27,78,36,87)(37,139,46,130)(38,138,47,129)(39,137,48,128)(40,136,49,127)(41,135,50,144)(42,134,51,143)(43,133,52,142)(44,132,53,141)(45,131,54,140)(55,120,64,111)(56,119,65,110)(57,118,66,109)(58,117,67,126)(59,116,68,125)(60,115,69,124)(61,114,70,123)(62,113,71,122)(63,112,72,121)>;
G:=Group( (1,54)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,55)(34,56)(35,57)(36,58)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136), (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)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(73,104)(74,105)(75,106)(76,107)(77,108)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137), (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), (1,94,10,103)(2,93,11,102)(3,92,12,101)(4,91,13,100)(5,108,14,99)(6,107,15,98)(7,106,16,97)(8,105,17,96)(9,104,18,95)(19,86,28,77)(20,85,29,76)(21,84,30,75)(22,83,31,74)(23,82,32,73)(24,81,33,90)(25,80,34,89)(26,79,35,88)(27,78,36,87)(37,139,46,130)(38,138,47,129)(39,137,48,128)(40,136,49,127)(41,135,50,144)(42,134,51,143)(43,133,52,142)(44,132,53,141)(45,131,54,140)(55,120,64,111)(56,119,65,110)(57,118,66,109)(58,117,67,126)(59,116,68,125)(60,115,69,124)(61,114,70,123)(62,113,71,122)(63,112,72,121) );
G=PermutationGroup([[(1,54),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,55),(34,56),(35,57),(36,58),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(106,134),(107,135),(108,136)], [(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),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(73,104),(74,105),(75,106),(76,107),(77,108),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(109,138),(110,139),(111,140),(112,141),(113,142),(114,143),(115,144),(116,127),(117,128),(118,129),(119,130),(120,131),(121,132),(122,133),(123,134),(124,135),(125,136),(126,137)], [(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)], [(1,94,10,103),(2,93,11,102),(3,92,12,101),(4,91,13,100),(5,108,14,99),(6,107,15,98),(7,106,16,97),(8,105,17,96),(9,104,18,95),(19,86,28,77),(20,85,29,76),(21,84,30,75),(22,83,31,74),(23,82,32,73),(24,81,33,90),(25,80,34,89),(26,79,35,88),(27,78,36,87),(37,139,46,130),(38,138,47,129),(39,137,48,128),(40,136,49,127),(41,135,50,144),(42,134,51,143),(43,133,52,142),(44,132,53,141),(45,131,54,140),(55,120,64,111),(56,119,65,110),(57,118,66,109),(58,117,67,126),(59,116,68,125),(60,115,69,124),(61,114,70,123),(62,113,71,122),(63,112,72,121)]])
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