metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D72⋊5C2, Q16⋊3D9, D18.3D4, C8.10D18, C24.13D6, C72.8C22, Q8.10D18, C36.10C23, Dic9.14D4, D36.5C22, (C8×D9)⋊3C2, C9⋊4(C4○D8), (C9×Q16)⋊3C2, C6.98(S3×D4), C2.24(D4×D9), C9⋊C8.8C22, Q8⋊2D9⋊4C2, Q8⋊3D9⋊3C2, C18.36(C2×D4), (C3×Q8).30D6, (C3×Q16).4S3, C3.(D24⋊C2), C4.10(C22×D9), (Q8×C9).5C22, C12.49(C22×S3), (C4×D9).12C22, SmallGroup(288,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D72⋊5C2
G = < a,b,c | a72=b2=c2=1, bab=a-1, cac=a17, cbc=a52b >
Subgroups: 488 in 93 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, C9, Dic3, C12, C12, D6, C2×C8, D8, SD16, Q16, C4○D4, D9, C18, C3⋊C8, C24, C4×S3, D12, C3×Q8, C4○D8, Dic9, C36, C36, D18, D18, S3×C8, D24, Q8⋊2S3, C3×Q16, Q8⋊3S3, C9⋊C8, C72, C4×D9, C4×D9, D36, D36, Q8×C9, D24⋊C2, C8×D9, D72, Q8⋊2D9, C9×Q16, Q8⋊3D9, D72⋊5C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, C4○D8, D18, S3×D4, C22×D9, D24⋊C2, D4×D9, D72⋊5C2
►On 144 points
Generators in S144
(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 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(73 134)(74 133)(75 132)(76 131)(77 130)(78 129)(79 128)(80 127)(81 126)(82 125)(83 124)(84 123)(85 122)(86 121)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(135 144)(136 143)(137 142)(138 141)(139 140)
(1 122)(2 139)(3 84)(4 101)(5 118)(6 135)(7 80)(8 97)(9 114)(10 131)(11 76)(12 93)(13 110)(14 127)(15 144)(16 89)(17 106)(18 123)(19 140)(20 85)(21 102)(22 119)(23 136)(24 81)(25 98)(26 115)(27 132)(28 77)(29 94)(30 111)(31 128)(32 73)(33 90)(34 107)(35 124)(36 141)(37 86)(38 103)(39 120)(40 137)(41 82)(42 99)(43 116)(44 133)(45 78)(46 95)(47 112)(48 129)(49 74)(50 91)(51 108)(52 125)(53 142)(54 87)(55 104)(56 121)(57 138)(58 83)(59 100)(60 117)(61 134)(62 79)(63 96)(64 113)(65 130)(66 75)(67 92)(68 109)(69 126)(70 143)(71 88)(72 105)
G:=sub<Sym(144)| (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(135,144)(136,143)(137,142)(138,141)(139,140), (1,122)(2,139)(3,84)(4,101)(5,118)(6,135)(7,80)(8,97)(9,114)(10,131)(11,76)(12,93)(13,110)(14,127)(15,144)(16,89)(17,106)(18,123)(19,140)(20,85)(21,102)(22,119)(23,136)(24,81)(25,98)(26,115)(27,132)(28,77)(29,94)(30,111)(31,128)(32,73)(33,90)(34,107)(35,124)(36,141)(37,86)(38,103)(39,120)(40,137)(41,82)(42,99)(43,116)(44,133)(45,78)(46,95)(47,112)(48,129)(49,74)(50,91)(51,108)(52,125)(53,142)(54,87)(55,104)(56,121)(57,138)(58,83)(59,100)(60,117)(61,134)(62,79)(63,96)(64,113)(65,130)(66,75)(67,92)(68,109)(69,126)(70,143)(71,88)(72,105)>;
G:=Group( (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(135,144)(136,143)(137,142)(138,141)(139,140), (1,122)(2,139)(3,84)(4,101)(5,118)(6,135)(7,80)(8,97)(9,114)(10,131)(11,76)(12,93)(13,110)(14,127)(15,144)(16,89)(17,106)(18,123)(19,140)(20,85)(21,102)(22,119)(23,136)(24,81)(25,98)(26,115)(27,132)(28,77)(29,94)(30,111)(31,128)(32,73)(33,90)(34,107)(35,124)(36,141)(37,86)(38,103)(39,120)(40,137)(41,82)(42,99)(43,116)(44,133)(45,78)(46,95)(47,112)(48,129)(49,74)(50,91)(51,108)(52,125)(53,142)(54,87)(55,104)(56,121)(57,138)(58,83)(59,100)(60,117)(61,134)(62,79)(63,96)(64,113)(65,130)(66,75)(67,92)(68,109)(69,126)(70,143)(71,88)(72,105) );
G=PermutationGroup([[(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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(73,134),(74,133),(75,132),(76,131),(77,130),(78,129),(79,128),(80,127),(81,126),(82,125),(83,124),(84,123),(85,122),(86,121),(87,120),(88,119),(89,118),(90,117),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,105),(103,104),(135,144),(136,143),(137,142),(138,141),(139,140)], [(1,122),(2,139),(3,84),(4,101),(5,118),(6,135),(7,80),(8,97),(9,114),(10,131),(11,76),(12,93),(13,110),(14,127),(15,144),(16,89),(17,106),(18,123),(19,140),(20,85),(21,102),(22,119),(23,136),(24,81),(25,98),(26,115),(27,132),(28,77),(29,94),(30,111),(31,128),(32,73),(33,90),(34,107),(35,124),(36,141),(37,86),(38,103),(39,120),(40,137),(41,82),(42,99),(43,116),(44,133),(45,78),(46,95),(47,112),(48,129),(49,74),(50,91),(51,108),(52,125),(53,142),(54,87),(55,104),(56,121),(57,138),(58,83),(59,100),(60,117),(61,134),(62,79),(63,96),(64,113),(65,130),(66,75),(67,92),(68,109),(69,126),(70,143),(71,88),(72,105)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 24A | 24B | 36A | 36B | 36C | 36D | ··· | 36I | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 36 | 36 | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 18 | 36 | 36 | 2 | 2 | 4 | 4 | 9 | 9 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | C4○D8 | D18 | D18 | S3×D4 | D24⋊C2 | D4×D9 | D72⋊5C2 |
| kernel | D72⋊5C2 | C8×D9 | D72 | Q8⋊2D9 | C9×Q16 | Q8⋊3D9 | C3×Q16 | Dic9 | D18 | C24 | C3×Q8 | Q16 | C9 | C8 | Q8 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of D72⋊5C2 ►in GL4(𝔽73) generated by
| 0 | 41 | 0 | 0 |
| 16 | 41 | 0 | 0 |
| 0 | 0 | 42 | 3 |
| 0 | 0 | 70 | 45 |
| 0 | 41 | 0 | 0 |
| 57 | 0 | 0 | 0 |
| 0 | 0 | 31 | 45 |
| 0 | 0 | 3 | 42 |
| 46 | 54 | 0 | 0 |
| 46 | 27 | 0 | 0 |
| 0 | 0 | 70 | 45 |
| 0 | 0 | 42 | 3 |
G:=sub<GL(4,GF(73))| [0,16,0,0,41,41,0,0,0,0,42,70,0,0,3,45],[0,57,0,0,41,0,0,0,0,0,31,3,0,0,45,42],[46,46,0,0,54,27,0,0,0,0,70,42,0,0,45,3] >;
D72⋊5C2 in GAP, Magma, Sage, TeX
D_{72}\rtimes_5C_2 % in TeX
G:=Group("D72:5C2"); // GroupNames label
G:=SmallGroup(288,129);
// by ID
G=gap.SmallGroup(288,129);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,100,346,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^72=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^17,c*b*c=a^52*b>;
// generators/relations