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, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C9, C12, D6, C2×C6, C2×C8, D8, C2×D4, D9, C18, C18, C24, D12, C2×C12, C22×S3, C2×D8, C36, D18, C2×C18, D24, C2×C24, C2×D12, C72, D36, D36, C2×C36, C22×D9, C2×D24, D72, C2×C72, C2×D36, C2×D72
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D9, D12, C22×S3, C2×D8, D18, D24, C2×D12, D36, C22×D9, C2×D24, D72, C2×D36, C2×D72
►On 144 points
Generators in S144
(1 137)(2 138)(3 139)(4 140)(5 141)(6 142)(7 143)(8 144)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 129)(66 130)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)
(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)(73 137)(74 136)(75 135)(76 134)(77 133)(78 132)(79 131)(80 130)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 120)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(138 144)(139 143)(140 142)
G:=sub<Sym(144)| (1,137)(2,138)(3,139)(4,140)(5,141)(6,142)(7,143)(8,144)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136), (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)(73,137)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,120)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(138,144)(139,143)(140,142)>;
G:=Group( (1,137)(2,138)(3,139)(4,140)(5,141)(6,142)(7,143)(8,144)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,129)(66,130)(67,131)(68,132)(69,133)(70,134)(71,135)(72,136), (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)(73,137)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,120)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(138,144)(139,143)(140,142) );
G=PermutationGroup([[(1,137),(2,138),(3,139),(4,140),(5,141),(6,142),(7,143),(8,144),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,121),(58,122),(59,123),(60,124),(61,125),(62,126),(63,127),(64,128),(65,129),(66,130),(67,131),(68,132),(69,133),(70,134),(71,135),(72,136)], [(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),(73,137),(74,136),(75,135),(76,134),(77,133),(78,132),(79,131),(80,130),(81,129),(82,128),(83,127),(84,126),(85,125),(86,124),(87,123),(88,122),(89,121),(90,120),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(138,144),(139,143),(140,142)]])
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