direct product, metabelian, supersoluble, monomial
Aliases: C22×C9⋊C12, C62.42D6, C62.11Dic3, C18⋊2(C2×C12), (C2×C18)⋊4C12, C9⋊2(C22×C12), (C2×Dic9)⋊6C6, Dic9⋊4(C2×C6), C23.4(C9⋊C6), (C2×C62).11S3, C18.9(C22×C6), (C22×C18).4C6, C6.23(C6×Dic3), (C22×Dic9)⋊3C3, C32.(C22×Dic3), 3- 1+2⋊2(C22×C4), (C22×3- 1+2)⋊2C4, (C2×3- 1+2).9C23, (C23×3- 1+2).2C2, (C22×3- 1+2).11C22, C6.47(S3×C2×C6), C3.3(Dic3×C2×C6), (C2×C6).65(S3×C6), C2.2(C22×C9⋊C6), (C2×C18).11(C2×C6), C22.11(C2×C9⋊C6), (C22×C6).31(C3×S3), (C3×C6).37(C22×S3), (C2×C6).23(C3×Dic3), (C3×C6).21(C2×Dic3), (C2×3- 1+2)⋊2(C2×C4), SmallGroup(432,378)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C22×C9⋊C12 |
Generators and relations for C22×C9⋊C12
G = < a,b,c,d | a2=b2=c9=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 446 in 178 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, 3- 1+2, Dic9, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C9⋊C12, C22×3- 1+2, C22×Dic9, Dic3×C2×C6, C2×C9⋊C12, C23×3- 1+2, C22×C9⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C9⋊C6, C6×Dic3, S3×C2×C6, C9⋊C12, C2×C9⋊C6, Dic3×C2×C6, C2×C9⋊C12, C22×C9⋊C6, C22×C9⋊C12
►On 144 points
Generators in S144
(1 30)(2 31)(3 32)(4 29)(5 15)(6 16)(7 13)(8 14)(9 39)(10 40)(11 37)(12 38)(17 27)(18 28)(19 25)(20 26)(21 47)(22 48)(23 45)(24 46)(33 42)(34 43)(35 44)(36 41)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 73)(60 74)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 97)(69 98)(70 99)(71 100)(72 101)(85 141)(86 142)(87 143)(88 144)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(109 131)(110 132)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 48)(2 45)(3 46)(4 47)(5 28)(6 25)(7 26)(8 27)(9 35)(10 36)(11 33)(12 34)(13 20)(14 17)(15 18)(16 19)(21 29)(22 30)(23 31)(24 32)(37 42)(38 43)(39 44)(40 41)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 97)(84 98)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)
(1 69 85 25 65 89 43 61 93)(2 90 70 44 86 62 26 94 66)(3 63 91 27 71 95 41 67 87)(4 96 64 42 92 68 28 88 72)(5 130 49 47 126 53 37 122 57)(6 54 131 38 50 123 48 58 127)(7 124 55 45 132 59 39 128 51)(8 60 125 40 56 129 46 52 121)(9 118 77 13 114 81 23 110 73)(10 82 119 24 78 111 14 74 115)(11 112 83 15 120 75 21 116 79)(12 76 113 22 84 117 16 80 109)(17 100 139 36 108 143 32 104 135)(18 144 101 29 140 105 33 136 97)(19 106 133 34 102 137 30 98 141)(20 138 107 31 134 99 35 142 103)
(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)
G:=sub<Sym(144)| (1,30)(2,31)(3,32)(4,29)(5,15)(6,16)(7,13)(8,14)(9,39)(10,40)(11,37)(12,38)(17,27)(18,28)(19,25)(20,26)(21,47)(22,48)(23,45)(24,46)(33,42)(34,43)(35,44)(36,41)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,73)(60,74)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,97)(69,98)(70,99)(71,100)(72,101)(85,141)(86,142)(87,143)(88,144)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(109,131)(110,132)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130), (1,48)(2,45)(3,46)(4,47)(5,28)(6,25)(7,26)(8,27)(9,35)(10,36)(11,33)(12,34)(13,20)(14,17)(15,18)(16,19)(21,29)(22,30)(23,31)(24,32)(37,42)(38,43)(39,44)(40,41)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,97)(84,98)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144), (1,69,85,25,65,89,43,61,93)(2,90,70,44,86,62,26,94,66)(3,63,91,27,71,95,41,67,87)(4,96,64,42,92,68,28,88,72)(5,130,49,47,126,53,37,122,57)(6,54,131,38,50,123,48,58,127)(7,124,55,45,132,59,39,128,51)(8,60,125,40,56,129,46,52,121)(9,118,77,13,114,81,23,110,73)(10,82,119,24,78,111,14,74,115)(11,112,83,15,120,75,21,116,79)(12,76,113,22,84,117,16,80,109)(17,100,139,36,108,143,32,104,135)(18,144,101,29,140,105,33,136,97)(19,106,133,34,102,137,30,98,141)(20,138,107,31,134,99,35,142,103), (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)>;
G:=Group( (1,30)(2,31)(3,32)(4,29)(5,15)(6,16)(7,13)(8,14)(9,39)(10,40)(11,37)(12,38)(17,27)(18,28)(19,25)(20,26)(21,47)(22,48)(23,45)(24,46)(33,42)(34,43)(35,44)(36,41)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,73)(60,74)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,97)(69,98)(70,99)(71,100)(72,101)(85,141)(86,142)(87,143)(88,144)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(109,131)(110,132)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130), (1,48)(2,45)(3,46)(4,47)(5,28)(6,25)(7,26)(8,27)(9,35)(10,36)(11,33)(12,34)(13,20)(14,17)(15,18)(16,19)(21,29)(22,30)(23,31)(24,32)(37,42)(38,43)(39,44)(40,41)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,97)(84,98)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144), (1,69,85,25,65,89,43,61,93)(2,90,70,44,86,62,26,94,66)(3,63,91,27,71,95,41,67,87)(4,96,64,42,92,68,28,88,72)(5,130,49,47,126,53,37,122,57)(6,54,131,38,50,123,48,58,127)(7,124,55,45,132,59,39,128,51)(8,60,125,40,56,129,46,52,121)(9,118,77,13,114,81,23,110,73)(10,82,119,24,78,111,14,74,115)(11,112,83,15,120,75,21,116,79)(12,76,113,22,84,117,16,80,109)(17,100,139,36,108,143,32,104,135)(18,144,101,29,140,105,33,136,97)(19,106,133,34,102,137,30,98,141)(20,138,107,31,134,99,35,142,103), (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) );
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,29),(5,15),(6,16),(7,13),(8,14),(9,39),(10,40),(11,37),(12,38),(17,27),(18,28),(19,25),(20,26),(21,47),(22,48),(23,45),(24,46),(33,42),(34,43),(35,44),(36,41),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,73),(60,74),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,97),(69,98),(70,99),(71,100),(72,101),(85,141),(86,142),(87,143),(88,144),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(109,131),(110,132),(111,121),(112,122),(113,123),(114,124),(115,125),(116,126),(117,127),(118,128),(119,129),(120,130)], [(1,48),(2,45),(3,46),(4,47),(5,28),(6,25),(7,26),(8,27),(9,35),(10,36),(11,33),(12,34),(13,20),(14,17),(15,18),(16,19),(21,29),(22,30),(23,31),(24,32),(37,42),(38,43),(39,44),(40,41),(49,72),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,97),(84,98),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,141),(118,142),(119,143),(120,144)], [(1,69,85,25,65,89,43,61,93),(2,90,70,44,86,62,26,94,66),(3,63,91,27,71,95,41,67,87),(4,96,64,42,92,68,28,88,72),(5,130,49,47,126,53,37,122,57),(6,54,131,38,50,123,48,58,127),(7,124,55,45,132,59,39,128,51),(8,60,125,40,56,129,46,52,121),(9,118,77,13,114,81,23,110,73),(10,82,119,24,78,111,14,74,115),(11,112,83,15,120,75,21,116,79),(12,76,113,22,84,117,16,80,109),(17,100,139,36,108,143,32,104,135),(18,144,101,29,140,105,33,136,97),(19,106,133,34,102,137,30,98,141),(20,138,107,31,134,99,35,142,103)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 4A | ··· | 4H | 6A | ··· | 6G | 6H | ··· | 6U | 9A | 9B | 9C | 12A | ··· | 12P | 18A | ··· | 18U |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | ··· | 1 | 2 | 3 | 3 | 9 | ··· | 9 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 9 | ··· | 9 | 6 | ··· | 6 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 | C9⋊C6 | C9⋊C12 | C2×C9⋊C6 |
| kernel | C22×C9⋊C12 | C2×C9⋊C12 | C23×3- 1+2 | C22×Dic9 | C22×3- 1+2 | C2×Dic9 | C22×C18 | C2×C18 | C2×C62 | C62 | C62 | C22×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 | 1 | 4 | 3 | 2 | 8 | 6 | 1 | 4 | 3 |
Matrix representation of C22×C9⋊C12 ►in GL10(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 36 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 29 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 33 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 14 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 | 23 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 7 | 0 | 0 |
G:=sub<GL(10,GF(37))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0],[29,33,0,0,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,0,0,29,33,0,0,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,0,0,30,14,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,30,14,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,30,0,0,0,0,0,0,0,0,7,23,0,0] >;
C22×C9⋊C12 in GAP, Magma, Sage, TeX
C_2^2\times C_9\rtimes C_{12} % in TeX
G:=Group("C2^2xC9:C12"); // GroupNames label
G:=SmallGroup(432,378);
// by ID
G=gap.SmallGroup(432,378);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,10085,1034,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^9=d^12=1,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^5>;
// generators/relations