metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D36⋊2C4, C2.2D72, C18.5D8, C6.5D24, C36.46D4, C18.3SD16, C22.10D36, (C2×C8)⋊2D9, (C2×C72)⋊2C2, C4.8(C4×D9), (C2×C24).4S3, C4⋊Dic9⋊1C2, C9⋊2(D4⋊C4), C12.56(C4×S3), C36.18(C2×C4), (C2×D36).1C2, (C2×C18).15D4, (C2×C4).76D18, (C2×C6).23D12, C3.(C2.D24), C6.3(C24⋊C2), C2.8(D18⋊C4), C6.14(D6⋊C4), (C2×C12).365D6, C2.3(C72⋊C2), C4.20(C9⋊D4), C18.7(C22⋊C4), (C2×C36).84C22, C12.107(C3⋊D4), SmallGroup(288,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D72
G = < a,b,c | a2=b72=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 488 in 75 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, C9, Dic3, C12, D6, C2×C6, C4⋊C4, C2×C8, C2×D4, D9, C18, C24, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, Dic9, C36, D18, C2×C18, C4⋊Dic3, C2×C24, C2×D12, C72, D36, D36, C2×Dic9, C2×C36, C22×D9, C2.D24, C4⋊Dic9, C2×C72, C2×D36, C2.D72
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, D9, C4×S3, D12, C3⋊D4, D4⋊C4, D18, C24⋊C2, D24, D6⋊C4, C4×D9, D36, C9⋊D4, C2.D24, C72⋊C2, D72, D18⋊C4, C2.D72
►On 144 points
Generators in S144
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 144)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)
(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 95 96 72)(2 71 97 94)(3 93 98 70)(4 69 99 92)(5 91 100 68)(6 67 101 90)(7 89 102 66)(8 65 103 88)(9 87 104 64)(10 63 105 86)(11 85 106 62)(12 61 107 84)(13 83 108 60)(14 59 109 82)(15 81 110 58)(16 57 111 80)(17 79 112 56)(18 55 113 78)(19 77 114 54)(20 53 115 76)(21 75 116 52)(22 51 117 74)(23 73 118 50)(24 49 119 144)(25 143 120 48)(26 47 121 142)(27 141 122 46)(28 45 123 140)(29 139 124 44)(30 43 125 138)(31 137 126 42)(32 41 127 136)(33 135 128 40)(34 39 129 134)(35 133 130 38)(36 37 131 132)
G:=sub<Sym(144)| (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95), (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), (1,95,96,72)(2,71,97,94)(3,93,98,70)(4,69,99,92)(5,91,100,68)(6,67,101,90)(7,89,102,66)(8,65,103,88)(9,87,104,64)(10,63,105,86)(11,85,106,62)(12,61,107,84)(13,83,108,60)(14,59,109,82)(15,81,110,58)(16,57,111,80)(17,79,112,56)(18,55,113,78)(19,77,114,54)(20,53,115,76)(21,75,116,52)(22,51,117,74)(23,73,118,50)(24,49,119,144)(25,143,120,48)(26,47,121,142)(27,141,122,46)(28,45,123,140)(29,139,124,44)(30,43,125,138)(31,137,126,42)(32,41,127,136)(33,135,128,40)(34,39,129,134)(35,133,130,38)(36,37,131,132)>;
G:=Group( (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,94)(72,95), (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), (1,95,96,72)(2,71,97,94)(3,93,98,70)(4,69,99,92)(5,91,100,68)(6,67,101,90)(7,89,102,66)(8,65,103,88)(9,87,104,64)(10,63,105,86)(11,85,106,62)(12,61,107,84)(13,83,108,60)(14,59,109,82)(15,81,110,58)(16,57,111,80)(17,79,112,56)(18,55,113,78)(19,77,114,54)(20,53,115,76)(21,75,116,52)(22,51,117,74)(23,73,118,50)(24,49,119,144)(25,143,120,48)(26,47,121,142)(27,141,122,46)(28,45,123,140)(29,139,124,44)(30,43,125,138)(31,137,126,42)(32,41,127,136)(33,135,128,40)(34,39,129,134)(35,133,130,38)(36,37,131,132) );
G=PermutationGroup([[(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,124),(30,125),(31,126),(32,127),(33,128),(34,129),(35,130),(36,131),(37,132),(38,133),(39,134),(40,135),(41,136),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,144),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,94),(72,95)], [(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)], [(1,95,96,72),(2,71,97,94),(3,93,98,70),(4,69,99,92),(5,91,100,68),(6,67,101,90),(7,89,102,66),(8,65,103,88),(9,87,104,64),(10,63,105,86),(11,85,106,62),(12,61,107,84),(13,83,108,60),(14,59,109,82),(15,81,110,58),(16,57,111,80),(17,79,112,56),(18,55,113,78),(19,77,114,54),(20,53,115,76),(21,75,116,52),(22,51,117,74),(23,73,118,50),(24,49,119,144),(25,143,120,48),(26,47,121,142),(27,141,122,46),(28,45,123,140),(29,139,124,44),(30,43,125,138),(31,137,126,42),(32,41,127,136),(33,135,128,40),(34,39,129,134),(35,133,130,38),(36,37,131,132)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 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 | 3 | 4 | 4 | 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 | 2 | 2 | 2 | 36 | 36 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | D9 | C4×S3 | C3⋊D4 | D12 | D18 | C24⋊C2 | D24 | C4×D9 | C9⋊D4 | D36 | C72⋊C2 | D72 |
| kernel | C2.D72 | C4⋊Dic9 | C2×C72 | C2×D36 | D36 | C2×C24 | C36 | C2×C18 | C2×C12 | C18 | C18 | C2×C8 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C2.D72 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 27 | 0 | 0 |
| 0 | 11 | 71 |
| 0 | 2 | 13 |
| 46 | 0 | 0 |
| 0 | 11 | 71 |
| 0 | 60 | 62 |
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[27,0,0,0,11,2,0,71,13],[46,0,0,0,11,60,0,71,62] >;
C2.D72 in GAP, Magma, Sage, TeX
C_2.D_{72} % in TeX
G:=Group("C2.D72"); // GroupNames label
G:=SmallGroup(288,28);
// by ID
G=gap.SmallGroup(288,28);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,422,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^2=b^72=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations