direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C42×D9, (C4×C36)⋊8C2, C36⋊5(C2×C4), C9⋊1(C2×C42), C3.(S3×C42), C12.71(C4×S3), (C4×C12).18S3, Dic9⋊5(C2×C4), D18.7(C2×C4), (C2×C4).96D18, (C4×Dic9)⋊17C2, (C2×C12).409D6, C18.2(C22×C4), (C2×C18).12C23, C22.9(C22×D9), (C2×C36).107C22, (C2×Dic9).47C22, (C22×D9).32C22, C2.1(C2×C4×D9), C6.41(S3×C2×C4), (C2×C4×D9).11C2, (C2×C6).169(C22×S3), SmallGroup(288,81)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C42×D9 |
Generators and relations for C42×D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 552 in 162 conjugacy classes, 84 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C42, C42, C22×C4, D9, C18, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C42, Dic9, C36, D18, C2×C18, C4×Dic3, C4×C12, S3×C2×C4, C4×D9, C2×Dic9, C2×C36, C22×D9, S3×C42, C4×Dic9, C4×C36, C2×C4×D9, C42×D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, D9, C4×S3, C22×S3, C2×C42, D18, S3×C2×C4, C4×D9, C22×D9, S3×C42, C2×C4×D9, C42×D9
►On 144 points
Generators in S144
(1 86 14 77)(2 87 15 78)(3 88 16 79)(4 89 17 80)(5 90 18 81)(6 82 10 73)(7 83 11 74)(8 84 12 75)(9 85 13 76)(19 100 28 91)(20 101 29 92)(21 102 30 93)(22 103 31 94)(23 104 32 95)(24 105 33 96)(25 106 34 97)(26 107 35 98)(27 108 36 99)(37 118 46 109)(38 119 47 110)(39 120 48 111)(40 121 49 112)(41 122 50 113)(42 123 51 114)(43 124 52 115)(44 125 53 116)(45 126 54 117)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 139 67 130)(59 140 68 131)(60 141 69 132)(61 142 70 133)(62 143 71 134)(63 144 72 135)
(1 68 23 50)(2 69 24 51)(3 70 25 52)(4 71 26 53)(5 72 27 54)(6 64 19 46)(7 65 20 47)(8 66 21 48)(9 67 22 49)(10 55 28 37)(11 56 29 38)(12 57 30 39)(13 58 31 40)(14 59 32 41)(15 60 33 42)(16 61 34 43)(17 62 35 44)(18 63 36 45)(73 136 91 118)(74 137 92 119)(75 138 93 120)(76 139 94 121)(77 140 95 122)(78 141 96 123)(79 142 97 124)(80 143 98 125)(81 144 99 126)(82 127 100 109)(83 128 101 110)(84 129 102 111)(85 130 103 112)(86 131 104 113)(87 132 105 114)(88 133 106 115)(89 134 107 116)(90 135 108 117)
(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 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 99)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 108)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 135)(118 143)(119 142)(120 141)(121 140)(122 139)(123 138)(124 137)(125 136)(126 144)
G:=sub<Sym(144)| (1,86,14,77)(2,87,15,78)(3,88,16,79)(4,89,17,80)(5,90,18,81)(6,82,10,73)(7,83,11,74)(8,84,12,75)(9,85,13,76)(19,100,28,91)(20,101,29,92)(21,102,30,93)(22,103,31,94)(23,104,32,95)(24,105,33,96)(25,106,34,97)(26,107,35,98)(27,108,36,99)(37,118,46,109)(38,119,47,110)(39,120,48,111)(40,121,49,112)(41,122,50,113)(42,123,51,114)(43,124,52,115)(44,125,53,116)(45,126,54,117)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,139,67,130)(59,140,68,131)(60,141,69,132)(61,142,70,133)(62,143,71,134)(63,144,72,135), (1,68,23,50)(2,69,24,51)(3,70,25,52)(4,71,26,53)(5,72,27,54)(6,64,19,46)(7,65,20,47)(8,66,21,48)(9,67,22,49)(10,55,28,37)(11,56,29,38)(12,57,30,39)(13,58,31,40)(14,59,32,41)(15,60,33,42)(16,61,34,43)(17,62,35,44)(18,63,36,45)(73,136,91,118)(74,137,92,119)(75,138,93,120)(76,139,94,121)(77,140,95,122)(78,141,96,123)(79,142,97,124)(80,143,98,125)(81,144,99,126)(82,127,100,109)(83,128,101,110)(84,129,102,111)(85,130,103,112)(86,131,104,113)(87,132,105,114)(88,133,106,115)(89,134,107,116)(90,135,108,117), (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), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,99)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,108)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,135)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,144)>;
G:=Group( (1,86,14,77)(2,87,15,78)(3,88,16,79)(4,89,17,80)(5,90,18,81)(6,82,10,73)(7,83,11,74)(8,84,12,75)(9,85,13,76)(19,100,28,91)(20,101,29,92)(21,102,30,93)(22,103,31,94)(23,104,32,95)(24,105,33,96)(25,106,34,97)(26,107,35,98)(27,108,36,99)(37,118,46,109)(38,119,47,110)(39,120,48,111)(40,121,49,112)(41,122,50,113)(42,123,51,114)(43,124,52,115)(44,125,53,116)(45,126,54,117)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,139,67,130)(59,140,68,131)(60,141,69,132)(61,142,70,133)(62,143,71,134)(63,144,72,135), (1,68,23,50)(2,69,24,51)(3,70,25,52)(4,71,26,53)(5,72,27,54)(6,64,19,46)(7,65,20,47)(8,66,21,48)(9,67,22,49)(10,55,28,37)(11,56,29,38)(12,57,30,39)(13,58,31,40)(14,59,32,41)(15,60,33,42)(16,61,34,43)(17,62,35,44)(18,63,36,45)(73,136,91,118)(74,137,92,119)(75,138,93,120)(76,139,94,121)(77,140,95,122)(78,141,96,123)(79,142,97,124)(80,143,98,125)(81,144,99,126)(82,127,100,109)(83,128,101,110)(84,129,102,111)(85,130,103,112)(86,131,104,113)(87,132,105,114)(88,133,106,115)(89,134,107,116)(90,135,108,117), (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), (1,22)(2,21)(3,20)(4,19)(5,27)(6,26)(7,25)(8,24)(9,23)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,99)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,108)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,135)(118,143)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)(125,136)(126,144) );
G=PermutationGroup([[(1,86,14,77),(2,87,15,78),(3,88,16,79),(4,89,17,80),(5,90,18,81),(6,82,10,73),(7,83,11,74),(8,84,12,75),(9,85,13,76),(19,100,28,91),(20,101,29,92),(21,102,30,93),(22,103,31,94),(23,104,32,95),(24,105,33,96),(25,106,34,97),(26,107,35,98),(27,108,36,99),(37,118,46,109),(38,119,47,110),(39,120,48,111),(40,121,49,112),(41,122,50,113),(42,123,51,114),(43,124,52,115),(44,125,53,116),(45,126,54,117),(55,136,64,127),(56,137,65,128),(57,138,66,129),(58,139,67,130),(59,140,68,131),(60,141,69,132),(61,142,70,133),(62,143,71,134),(63,144,72,135)], [(1,68,23,50),(2,69,24,51),(3,70,25,52),(4,71,26,53),(5,72,27,54),(6,64,19,46),(7,65,20,47),(8,66,21,48),(9,67,22,49),(10,55,28,37),(11,56,29,38),(12,57,30,39),(13,58,31,40),(14,59,32,41),(15,60,33,42),(16,61,34,43),(17,62,35,44),(18,63,36,45),(73,136,91,118),(74,137,92,119),(75,138,93,120),(76,139,94,121),(77,140,95,122),(78,141,96,123),(79,142,97,124),(80,143,98,125),(81,144,99,126),(82,127,100,109),(83,128,101,110),(84,129,102,111),(85,130,103,112),(86,131,104,113),(87,132,105,114),(88,133,106,115),(89,134,107,116),(90,135,108,117)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,27),(6,26),(7,25),(8,24),(9,23),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,36),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,99),(82,107),(83,106),(84,105),(85,104),(86,103),(87,102),(88,101),(89,100),(90,108),(109,134),(110,133),(111,132),(112,131),(113,130),(114,129),(115,128),(116,127),(117,135),(118,143),(119,142),(120,141),(121,140),(122,139),(123,138),(124,137),(125,136),(126,144)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4X | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D9 | C4×S3 | D18 | C4×D9 |
| kernel | C42×D9 | C4×Dic9 | C4×C36 | C2×C4×D9 | C4×D9 | C4×C12 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 24 | 1 | 3 | 3 | 12 | 9 | 36 |
Matrix representation of C42×D9 ►in GL3(𝔽37) generated by
| 31 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 31 |
| 31 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 31 | 20 |
| 0 | 17 | 11 |
| 1 | 0 | 0 |
| 0 | 31 | 20 |
| 0 | 26 | 6 |
G:=sub<GL(3,GF(37))| [31,0,0,0,31,0,0,0,31],[31,0,0,0,36,0,0,0,36],[1,0,0,0,31,17,0,20,11],[1,0,0,0,31,26,0,20,6] >;
C42×D9 in GAP, Magma, Sage, TeX
C_4^2\times D_9
% in TeX
G:=Group("C4^2xD9"); // GroupNames label
G:=SmallGroup(288,81);
// by ID
G=gap.SmallGroup(288,81);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=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