direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D36, C36⋊2C23, D18⋊1C23, C18.3C24, C23.40D18, (C2×C18)⋊6D4, (C2×C4)⋊9D18, C18⋊1(C2×D4), C9⋊1(C22×D4), (C22×C4)⋊7D9, C4⋊2(C22×D9), C3.(C22×D12), (C22×C36)⋊7C2, C6.47(C2×D12), (C2×C6).37D12, (C23×D9)⋊3C2, C2.4(C23×D9), (C2×C36)⋊12C22, (C2×C12).380D6, C6.40(S3×C23), (C22×C12).22S3, (C2×C18).64C23, (C22×C6).148D6, (C22×D9)⋊5C22, C12.186(C22×S3), C22.30(C22×D9), (C22×C18).45C22, (C2×C6).221(C22×S3), SmallGroup(288,354)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×D36
G = < a,b,c,d | a2=b2=c36=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1784 in 354 conjugacy classes, 132 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], C9, C12 [×4], D6 [×32], C2×C6 [×7], C22×C4, C2×D4 [×12], C24 [×2], D9 [×8], C18, C18 [×6], D12 [×16], C2×C12 [×6], C22×S3 [×20], C22×C6, C22×D4, C36 [×4], D18 [×8], D18 [×24], C2×C18 [×7], C2×D12 [×12], C22×C12, S3×C23 [×2], D36 [×16], C2×C36 [×6], C22×D9 [×12], C22×D9 [×8], C22×C18, C22×D12, C2×D36 [×12], C22×C36, C23×D9 [×2], C22×D36
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D9, D12 [×4], C22×S3 [×7], C22×D4, D18 [×7], C2×D12 [×6], S3×C23, D36 [×4], C22×D9 [×7], C22×D12, C2×D36 [×6], C23×D9, C22×D36
►On 144 points
Generators in S144
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 121)(80 122)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 109)(104 110)(105 111)(106 112)(107 113)(108 114)
(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 9)(2 8)(3 7)(4 6)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(68 72)(69 71)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(106 108)(109 117)(110 116)(111 115)(112 114)(118 144)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)
G:=sub<Sym(144)| (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,121)(80,122)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(99,141)(100,142)(101,143)(102,144)(103,109)(104,110)(105,111)(106,112)(107,113)(108,114), (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), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(68,72)(69,71)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(106,108)(109,117)(110,116)(111,115)(112,114)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)>;
G:=Group( (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,121)(80,122)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(99,141)(100,142)(101,143)(102,144)(103,109)(104,110)(105,111)(106,112)(107,113)(108,114), (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), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(68,72)(69,71)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(106,108)(109,117)(110,116)(111,115)(112,114)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132) );
G=PermutationGroup([(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,109),(20,110),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,121),(80,122),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,133),(92,134),(93,135),(94,136),(95,137),(96,138),(97,139),(98,140),(99,141),(100,142),(101,143),(102,144),(103,109),(104,110),(105,111),(106,112),(107,113),(108,114)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(68,72),(69,71),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(106,108),(109,117),(110,116),(111,115),(112,114),(118,144),(119,143),(120,142),(121,141),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132)])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3 | 4A | 4B | 4C | 4D | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D9 | D12 | D18 | D18 | D36 |
| kernel | C22×D36 | C2×D36 | C22×C36 | C23×D9 | C22×C12 | C2×C18 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 1 | 4 | 6 | 1 | 3 | 8 | 18 | 3 | 24 |
Matrix representation of C22×D36 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 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 | 32 | 5 | 0 | 0 |
| 0 | 0 | 32 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 31 |
| 0 | 0 | 0 | 0 | 6 | 20 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 20 |
| 0 | 0 | 0 | 0 | 26 | 31 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,32,0,0,0,0,5,27,0,0,0,0,0,0,26,6,0,0,0,0,31,20],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,36,1,0,0,0,0,0,0,6,26,0,0,0,0,20,31] >;
C22×D36 in GAP, Magma, Sage, TeX
C_2^2\times D_{36} % in TeX
G:=Group("C2^2xD36"); // GroupNames label
G:=SmallGroup(288,354);
// by ID
G=gap.SmallGroup(288,354);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations