direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D72, C18⋊1D8, C8⋊7D18, C4.8D36, C6.6D24, C72⋊8C22, C24.71D6, C36.31D4, C12.42D12, D36⋊3C22, C36.30C23, C22.14D36, C9⋊1(C2×D8), (C2×C8)⋊3D9, C3.(C2×D24), (C2×C72)⋊5C2, (C2×D36)⋊5C2, (C2×C24).11S3, C6.40(C2×D12), C18.11(C2×D4), C2.13(C2×D36), (C2×C6).27D12, (C2×C18).18D4, (C2×C4).82D18, (C2×C12).372D6, C4.28(C22×D9), (C2×C36).91C22, C12.181(C22×S3), SmallGroup(288,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D72
G = < a,b,c | a2=b72=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 800 in 114 conjugacy classes, 44 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, D4 [×6], C23 [×2], C9, C12 [×2], D6 [×8], C2×C6, C2×C8, D8 [×4], C2×D4 [×2], D9 [×4], C18, C18 [×2], C24 [×2], D12 [×6], C2×C12, C22×S3 [×2], C2×D8, C36 [×2], D18 [×8], C2×C18, D24 [×4], C2×C24, C2×D12 [×2], C72 [×2], D36 [×4], D36 [×2], C2×C36, C22×D9 [×2], C2×D24, D72 [×4], C2×C72, C2×D36 [×2], C2×D72
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, D9, D12 [×2], C22×S3, C2×D8, D18 [×3], D24 [×2], C2×D12, D36 [×2], C22×D9, C2×D24, D72 [×2], C2×D36, C2×D72
►On 144 points
Generators in S144
(1 140)(2 141)(3 142)(4 143)(5 144)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)
(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 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(73 143)(74 142)(75 141)(76 140)(77 139)(78 138)(79 137)(80 136)(81 135)(82 134)(83 133)(84 132)(85 131)(86 130)(87 129)(88 128)(89 127)(90 126)(91 125)(92 124)(93 123)(94 122)(95 121)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
G:=sub<Sym(144)| (1,140)(2,141)(3,142)(4,143)(5,144)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139), (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)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(73,143)(74,142)(75,141)(76,140)(77,139)(78,138)(79,137)(80,136)(81,135)(82,134)(83,133)(84,132)(85,131)(86,130)(87,129)(88,128)(89,127)(90,126)(91,125)(92,124)(93,123)(94,122)(95,121)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)>;
G:=Group( (1,140)(2,141)(3,142)(4,143)(5,144)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139), (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)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(73,143)(74,142)(75,141)(76,140)(77,139)(78,138)(79,137)(80,136)(81,135)(82,134)(83,133)(84,132)(85,131)(86,130)(87,129)(88,128)(89,127)(90,126)(91,125)(92,124)(93,123)(94,122)(95,121)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109) );
G=PermutationGroup([(1,140),(2,141),(3,142),(4,143),(5,144),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139)], [(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),(10,72),(11,71),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(73,143),(74,142),(75,141),(76,140),(77,139),(78,138),(79,137),(80,136),(81,135),(82,134),(83,133),(84,132),(85,131),(86,130),(87,129),(88,128),(89,127),(90,126),(91,125),(92,124),(93,123),(94,122),(95,121),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109)])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D9 | D12 | D12 | D18 | D18 | D24 | D36 | D36 | D72 |
| kernel | C2×D72 | D72 | C2×C72 | C2×D36 | C2×C24 | C36 | C2×C18 | C24 | C2×C12 | C18 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C2×D72 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 44 | 0 | 0 |
| 29 | 54 | 0 | 0 |
| 0 | 0 | 63 | 41 |
| 0 | 0 | 32 | 22 |
| 3 | 31 | 0 | 0 |
| 28 | 70 | 0 | 0 |
| 0 | 0 | 3 | 31 |
| 0 | 0 | 28 | 70 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[25,29,0,0,44,54,0,0,0,0,63,32,0,0,41,22],[3,28,0,0,31,70,0,0,0,0,3,28,0,0,31,70] >;
C2×D72 in GAP, Magma, Sage, TeX
C_2\times D_{72} % in TeX
G:=Group("C2xD72"); // GroupNames label
G:=SmallGroup(288,114);
// by ID
G=gap.SmallGroup(288,114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,142,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^2=b^72=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations