metabelian, supersoluble, monomial
Aliases: C48⋊1S3, C3⋊1D48, C6.7D24, C32⋊5D16, C24.72D6, C12.43D12, (C3×C48)⋊1C2, C16⋊1(C3⋊S3), (C3×C6).23D8, C32⋊5D8⋊1C2, (C3×C12).118D4, C4.1(C12⋊S3), C2.3(C32⋊5D8), (C3×C24).50C22, C8.13(C2×C3⋊S3), SmallGroup(288,274)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊5D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 712 in 84 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C8, D4, C32, C12, D6, C16, D8, C3⋊S3, C3×C6, C24, D12, D16, C3×C12, C2×C3⋊S3, C48, D24, C3×C24, C12⋊S3, D48, C3×C48, C32⋊5D8, C32⋊5D16
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, D16, C2×C3⋊S3, D24, C12⋊S3, D48, C32⋊5D8, C32⋊5D16
►On 144 points
Generators in S144
(1 126 39)(2 127 40)(3 128 41)(4 113 42)(5 114 43)(6 115 44)(7 116 45)(8 117 46)(9 118 47)(10 119 48)(11 120 33)(12 121 34)(13 122 35)(14 123 36)(15 124 37)(16 125 38)(17 105 58)(18 106 59)(19 107 60)(20 108 61)(21 109 62)(22 110 63)(23 111 64)(24 112 49)(25 97 50)(26 98 51)(27 99 52)(28 100 53)(29 101 54)(30 102 55)(31 103 56)(32 104 57)(65 87 141)(66 88 142)(67 89 143)(68 90 144)(69 91 129)(70 92 130)(71 93 131)(72 94 132)(73 95 133)(74 96 134)(75 81 135)(76 82 136)(77 83 137)(78 84 138)(79 85 139)(80 86 140)
(1 87 22)(2 88 23)(3 89 24)(4 90 25)(5 91 26)(6 92 27)(7 93 28)(8 94 29)(9 95 30)(10 96 31)(11 81 32)(12 82 17)(13 83 18)(14 84 19)(15 85 20)(16 86 21)(33 75 57)(34 76 58)(35 77 59)(36 78 60)(37 79 61)(38 80 62)(39 65 63)(40 66 64)(41 67 49)(42 68 50)(43 69 51)(44 70 52)(45 71 53)(46 72 54)(47 73 55)(48 74 56)(97 113 144)(98 114 129)(99 115 130)(100 116 131)(101 117 132)(102 118 133)(103 119 134)(104 120 135)(105 121 136)(106 122 137)(107 123 138)(108 124 139)(109 125 140)(110 126 141)(111 127 142)(112 128 143)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 84)(25 83)(26 82)(27 81)(28 96)(29 95)(30 94)(31 93)(32 92)(33 115)(34 114)(35 113)(36 128)(37 127)(38 126)(39 125)(40 124)(41 123)(42 122)(43 121)(44 120)(45 119)(46 118)(47 117)(48 116)(49 138)(50 137)(51 136)(52 135)(53 134)(54 133)(55 132)(56 131)(57 130)(58 129)(59 144)(60 143)(61 142)(62 141)(63 140)(64 139)(65 109)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 112)(79 111)(80 110)
G:=sub<Sym(144)| (1,126,39)(2,127,40)(3,128,41)(4,113,42)(5,114,43)(6,115,44)(7,116,45)(8,117,46)(9,118,47)(10,119,48)(11,120,33)(12,121,34)(13,122,35)(14,123,36)(15,124,37)(16,125,38)(17,105,58)(18,106,59)(19,107,60)(20,108,61)(21,109,62)(22,110,63)(23,111,64)(24,112,49)(25,97,50)(26,98,51)(27,99,52)(28,100,53)(29,101,54)(30,102,55)(31,103,56)(32,104,57)(65,87,141)(66,88,142)(67,89,143)(68,90,144)(69,91,129)(70,92,130)(71,93,131)(72,94,132)(73,95,133)(74,96,134)(75,81,135)(76,82,136)(77,83,137)(78,84,138)(79,85,139)(80,86,140), (1,87,22)(2,88,23)(3,89,24)(4,90,25)(5,91,26)(6,92,27)(7,93,28)(8,94,29)(9,95,30)(10,96,31)(11,81,32)(12,82,17)(13,83,18)(14,84,19)(15,85,20)(16,86,21)(33,75,57)(34,76,58)(35,77,59)(36,78,60)(37,79,61)(38,80,62)(39,65,63)(40,66,64)(41,67,49)(42,68,50)(43,69,51)(44,70,52)(45,71,53)(46,72,54)(47,73,55)(48,74,56)(97,113,144)(98,114,129)(99,115,130)(100,116,131)(101,117,132)(102,118,133)(103,119,134)(104,120,135)(105,121,136)(106,122,137)(107,123,138)(108,124,139)(109,125,140)(110,126,141)(111,127,142)(112,128,143), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,96)(29,95)(30,94)(31,93)(32,92)(33,115)(34,114)(35,113)(36,128)(37,127)(38,126)(39,125)(40,124)(41,123)(42,122)(43,121)(44,120)(45,119)(46,118)(47,117)(48,116)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,112)(79,111)(80,110)>;
G:=Group( (1,126,39)(2,127,40)(3,128,41)(4,113,42)(5,114,43)(6,115,44)(7,116,45)(8,117,46)(9,118,47)(10,119,48)(11,120,33)(12,121,34)(13,122,35)(14,123,36)(15,124,37)(16,125,38)(17,105,58)(18,106,59)(19,107,60)(20,108,61)(21,109,62)(22,110,63)(23,111,64)(24,112,49)(25,97,50)(26,98,51)(27,99,52)(28,100,53)(29,101,54)(30,102,55)(31,103,56)(32,104,57)(65,87,141)(66,88,142)(67,89,143)(68,90,144)(69,91,129)(70,92,130)(71,93,131)(72,94,132)(73,95,133)(74,96,134)(75,81,135)(76,82,136)(77,83,137)(78,84,138)(79,85,139)(80,86,140), (1,87,22)(2,88,23)(3,89,24)(4,90,25)(5,91,26)(6,92,27)(7,93,28)(8,94,29)(9,95,30)(10,96,31)(11,81,32)(12,82,17)(13,83,18)(14,84,19)(15,85,20)(16,86,21)(33,75,57)(34,76,58)(35,77,59)(36,78,60)(37,79,61)(38,80,62)(39,65,63)(40,66,64)(41,67,49)(42,68,50)(43,69,51)(44,70,52)(45,71,53)(46,72,54)(47,73,55)(48,74,56)(97,113,144)(98,114,129)(99,115,130)(100,116,131)(101,117,132)(102,118,133)(103,119,134)(104,120,135)(105,121,136)(106,122,137)(107,123,138)(108,124,139)(109,125,140)(110,126,141)(111,127,142)(112,128,143), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,91)(18,90)(19,89)(20,88)(21,87)(22,86)(23,85)(24,84)(25,83)(26,82)(27,81)(28,96)(29,95)(30,94)(31,93)(32,92)(33,115)(34,114)(35,113)(36,128)(37,127)(38,126)(39,125)(40,124)(41,123)(42,122)(43,121)(44,120)(45,119)(46,118)(47,117)(48,116)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,112)(79,111)(80,110) );
G=PermutationGroup([[(1,126,39),(2,127,40),(3,128,41),(4,113,42),(5,114,43),(6,115,44),(7,116,45),(8,117,46),(9,118,47),(10,119,48),(11,120,33),(12,121,34),(13,122,35),(14,123,36),(15,124,37),(16,125,38),(17,105,58),(18,106,59),(19,107,60),(20,108,61),(21,109,62),(22,110,63),(23,111,64),(24,112,49),(25,97,50),(26,98,51),(27,99,52),(28,100,53),(29,101,54),(30,102,55),(31,103,56),(32,104,57),(65,87,141),(66,88,142),(67,89,143),(68,90,144),(69,91,129),(70,92,130),(71,93,131),(72,94,132),(73,95,133),(74,96,134),(75,81,135),(76,82,136),(77,83,137),(78,84,138),(79,85,139),(80,86,140)], [(1,87,22),(2,88,23),(3,89,24),(4,90,25),(5,91,26),(6,92,27),(7,93,28),(8,94,29),(9,95,30),(10,96,31),(11,81,32),(12,82,17),(13,83,18),(14,84,19),(15,85,20),(16,86,21),(33,75,57),(34,76,58),(35,77,59),(36,78,60),(37,79,61),(38,80,62),(39,65,63),(40,66,64),(41,67,49),(42,68,50),(43,69,51),(44,70,52),(45,71,53),(46,72,54),(47,73,55),(48,74,56),(97,113,144),(98,114,129),(99,115,130),(100,116,131),(101,117,132),(102,118,133),(103,119,134),(104,120,135),(105,121,136),(106,122,137),(107,123,138),(108,124,139),(109,125,140),(110,126,141),(111,127,142),(112,128,143)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,91),(18,90),(19,89),(20,88),(21,87),(22,86),(23,85),(24,84),(25,83),(26,82),(27,81),(28,96),(29,95),(30,94),(31,93),(32,92),(33,115),(34,114),(35,113),(36,128),(37,127),(38,126),(39,125),(40,124),(41,123),(42,122),(43,121),(44,120),(45,119),(46,118),(47,117),(48,116),(49,138),(50,137),(51,136),(52,135),(53,134),(54,133),(55,132),(56,131),(57,130),(58,129),(59,144),(60,143),(61,142),(62,141),(63,140),(64,139),(65,109),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,98),(77,97),(78,112),(79,111),(80,110)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 8A | 8B | 12A | ··· | 12H | 16A | 16B | 16C | 16D | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 72 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | D12 | D16 | D24 | D48 |
| kernel | C32⋊5D16 | C3×C48 | C32⋊5D8 | C48 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C32⋊5D16 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 96 | 96 |
| 0 | 1 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 96 | 96 |
| 65 | 78 | 0 | 0 |
| 19 | 84 | 0 | 0 |
| 0 | 0 | 68 | 39 |
| 0 | 0 | 58 | 29 |
| 65 | 78 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 29 | 68 |
| 0 | 0 | 39 | 68 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,0,96,0,0,1,96],[0,96,0,0,1,96,0,0,0,0,0,96,0,0,1,96],[65,19,0,0,78,84,0,0,0,0,68,58,0,0,39,29],[65,13,0,0,78,32,0,0,0,0,29,39,0,0,68,68] >;
C32⋊5D16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_5D_{16} % in TeX
G:=Group("C3^2:5D16"); // GroupNames label
G:=SmallGroup(288,274);
// by ID
G=gap.SmallGroup(288,274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,254,142,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations