metabelian, supersoluble, monomial
Aliases: C32⋊7D16, C24.18D6, (C3×D8)⋊1S3, D8⋊1(C3⋊S3), (C3×C6).37D8, C32⋊5D8⋊5C2, C3⋊3(C3⋊D16), (C3×C12).52D4, (C32×D8)⋊2C2, C24.S3⋊3C2, C6.23(D4⋊S3), C12.34(C3⋊D4), (C3×C24).17C22, C4.1(C32⋊7D4), C2.4(C32⋊7D8), C8.4(C2×C3⋊S3), SmallGroup(288,301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊7D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 488 in 84 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, D6, C2×C6, C16, D8, D8, C3⋊S3, C3×C6, C3×C6, C24, D12, C3×D4, D16, C3×C12, C2×C3⋊S3, C62, C3⋊C16, D24, C3×D8, C3×C24, C12⋊S3, D4×C32, C3⋊D16, C24.S3, C32⋊5D8, C32×D8, C32⋊7D16
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, C3⋊D4, D16, C2×C3⋊S3, D4⋊S3, C32⋊7D4, C3⋊D16, C32⋊7D8, C32⋊7D16
►On 144 points
Generators in S144
(1 113 61)(2 62 114)(3 115 63)(4 64 116)(5 117 49)(6 50 118)(7 119 51)(8 52 120)(9 121 53)(10 54 122)(11 123 55)(12 56 124)(13 125 57)(14 58 126)(15 127 59)(16 60 128)(17 102 73)(18 74 103)(19 104 75)(20 76 105)(21 106 77)(22 78 107)(23 108 79)(24 80 109)(25 110 65)(26 66 111)(27 112 67)(28 68 97)(29 98 69)(30 70 99)(31 100 71)(32 72 101)(33 87 132)(34 133 88)(35 89 134)(36 135 90)(37 91 136)(38 137 92)(39 93 138)(40 139 94)(41 95 140)(42 141 96)(43 81 142)(44 143 82)(45 83 144)(46 129 84)(47 85 130)(48 131 86)
(1 134 76)(2 77 135)(3 136 78)(4 79 137)(5 138 80)(6 65 139)(7 140 66)(8 67 141)(9 142 68)(10 69 143)(11 144 70)(12 71 129)(13 130 72)(14 73 131)(15 132 74)(16 75 133)(17 86 58)(18 59 87)(19 88 60)(20 61 89)(21 90 62)(22 63 91)(23 92 64)(24 49 93)(25 94 50)(26 51 95)(27 96 52)(28 53 81)(29 82 54)(30 55 83)(31 84 56)(32 57 85)(33 103 127)(34 128 104)(35 105 113)(36 114 106)(37 107 115)(38 116 108)(39 109 117)(40 118 110)(41 111 119)(42 120 112)(43 97 121)(44 122 98)(45 99 123)(46 124 100)(47 101 125)(48 126 102)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(49 125)(50 124)(51 123)(52 122)(53 121)(54 120)(55 119)(56 118)(57 117)(58 116)(59 115)(60 114)(61 113)(62 128)(63 127)(64 126)(65 129)(66 144)(67 143)(68 142)(69 141)(70 140)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 133)(78 132)(79 131)(80 130)(81 97)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)
G:=sub<Sym(144)| (1,113,61)(2,62,114)(3,115,63)(4,64,116)(5,117,49)(6,50,118)(7,119,51)(8,52,120)(9,121,53)(10,54,122)(11,123,55)(12,56,124)(13,125,57)(14,58,126)(15,127,59)(16,60,128)(17,102,73)(18,74,103)(19,104,75)(20,76,105)(21,106,77)(22,78,107)(23,108,79)(24,80,109)(25,110,65)(26,66,111)(27,112,67)(28,68,97)(29,98,69)(30,70,99)(31,100,71)(32,72,101)(33,87,132)(34,133,88)(35,89,134)(36,135,90)(37,91,136)(38,137,92)(39,93,138)(40,139,94)(41,95,140)(42,141,96)(43,81,142)(44,143,82)(45,83,144)(46,129,84)(47,85,130)(48,131,86), (1,134,76)(2,77,135)(3,136,78)(4,79,137)(5,138,80)(6,65,139)(7,140,66)(8,67,141)(9,142,68)(10,69,143)(11,144,70)(12,71,129)(13,130,72)(14,73,131)(15,132,74)(16,75,133)(17,86,58)(18,59,87)(19,88,60)(20,61,89)(21,90,62)(22,63,91)(23,92,64)(24,49,93)(25,94,50)(26,51,95)(27,96,52)(28,53,81)(29,82,54)(30,55,83)(31,84,56)(32,57,85)(33,103,127)(34,128,104)(35,105,113)(36,114,106)(37,107,115)(38,116,108)(39,109,117)(40,118,110)(41,111,119)(42,120,112)(43,97,121)(44,122,98)(45,99,123)(46,124,100)(47,101,125)(48,126,102), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(49,125)(50,124)(51,123)(52,122)(53,121)(54,120)(55,119)(56,118)(57,117)(58,116)(59,115)(60,114)(61,113)(62,128)(63,127)(64,126)(65,129)(66,144)(67,143)(68,142)(69,141)(70,140)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130)(81,97)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)>;
G:=Group( (1,113,61)(2,62,114)(3,115,63)(4,64,116)(5,117,49)(6,50,118)(7,119,51)(8,52,120)(9,121,53)(10,54,122)(11,123,55)(12,56,124)(13,125,57)(14,58,126)(15,127,59)(16,60,128)(17,102,73)(18,74,103)(19,104,75)(20,76,105)(21,106,77)(22,78,107)(23,108,79)(24,80,109)(25,110,65)(26,66,111)(27,112,67)(28,68,97)(29,98,69)(30,70,99)(31,100,71)(32,72,101)(33,87,132)(34,133,88)(35,89,134)(36,135,90)(37,91,136)(38,137,92)(39,93,138)(40,139,94)(41,95,140)(42,141,96)(43,81,142)(44,143,82)(45,83,144)(46,129,84)(47,85,130)(48,131,86), (1,134,76)(2,77,135)(3,136,78)(4,79,137)(5,138,80)(6,65,139)(7,140,66)(8,67,141)(9,142,68)(10,69,143)(11,144,70)(12,71,129)(13,130,72)(14,73,131)(15,132,74)(16,75,133)(17,86,58)(18,59,87)(19,88,60)(20,61,89)(21,90,62)(22,63,91)(23,92,64)(24,49,93)(25,94,50)(26,51,95)(27,96,52)(28,53,81)(29,82,54)(30,55,83)(31,84,56)(32,57,85)(33,103,127)(34,128,104)(35,105,113)(36,114,106)(37,107,115)(38,116,108)(39,109,117)(40,118,110)(41,111,119)(42,120,112)(43,97,121)(44,122,98)(45,99,123)(46,124,100)(47,101,125)(48,126,102), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(49,125)(50,124)(51,123)(52,122)(53,121)(54,120)(55,119)(56,118)(57,117)(58,116)(59,115)(60,114)(61,113)(62,128)(63,127)(64,126)(65,129)(66,144)(67,143)(68,142)(69,141)(70,140)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130)(81,97)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98) );
G=PermutationGroup([[(1,113,61),(2,62,114),(3,115,63),(4,64,116),(5,117,49),(6,50,118),(7,119,51),(8,52,120),(9,121,53),(10,54,122),(11,123,55),(12,56,124),(13,125,57),(14,58,126),(15,127,59),(16,60,128),(17,102,73),(18,74,103),(19,104,75),(20,76,105),(21,106,77),(22,78,107),(23,108,79),(24,80,109),(25,110,65),(26,66,111),(27,112,67),(28,68,97),(29,98,69),(30,70,99),(31,100,71),(32,72,101),(33,87,132),(34,133,88),(35,89,134),(36,135,90),(37,91,136),(38,137,92),(39,93,138),(40,139,94),(41,95,140),(42,141,96),(43,81,142),(44,143,82),(45,83,144),(46,129,84),(47,85,130),(48,131,86)], [(1,134,76),(2,77,135),(3,136,78),(4,79,137),(5,138,80),(6,65,139),(7,140,66),(8,67,141),(9,142,68),(10,69,143),(11,144,70),(12,71,129),(13,130,72),(14,73,131),(15,132,74),(16,75,133),(17,86,58),(18,59,87),(19,88,60),(20,61,89),(21,90,62),(22,63,91),(23,92,64),(24,49,93),(25,94,50),(26,51,95),(27,96,52),(28,53,81),(29,82,54),(30,55,83),(31,84,56),(32,57,85),(33,103,127),(34,128,104),(35,105,113),(36,114,106),(37,107,115),(38,116,108),(39,109,117),(40,118,110),(41,111,119),(42,120,112),(43,97,121),(44,122,98),(45,99,123),(46,124,100),(47,101,125),(48,126,102)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(49,125),(50,124),(51,123),(52,122),(53,121),(54,120),(55,119),(56,118),(57,117),(58,116),(59,115),(60,114),(61,113),(62,128),(63,127),(64,126),(65,129),(66,144),(67,143),(68,142),(69,141),(70,140),(71,139),(72,138),(73,137),(74,136),(75,135),(76,134),(77,133),(78,132),(79,131),(80,130),(81,97),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 8 | 72 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D8 | C3⋊D4 | D16 | D4⋊S3 | C3⋊D16 |
| kernel | C32⋊7D16 | C24.S3 | C32⋊5D8 | C32×D8 | C3×D8 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 4 | 4 | 8 |
Matrix representation of C32⋊7D16 ►in GL6(𝔽97)
| 0 | 96 | 0 | 0 | 0 | 0 |
| 1 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 96 | 1 | 0 | 0 | 0 | 0 |
| 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 1 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 41 | 0 | 0 | 0 | 0 |
| 56 | 82 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 52 |
| 0 | 0 | 0 | 0 | 71 | 69 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 95 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(97))| [0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,56,0,0,0,0,41,82,0,0,0,0,0,0,0,96,0,0,0,0,96,0,0,0,0,0,0,0,24,71,0,0,0,0,52,69],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,96,0,0,0,0,0,95,1] >;
C32⋊7D16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_7D_{16} % in TeX
G:=Group("C3^2:7D16"); // GroupNames label
G:=SmallGroup(288,301);
// by ID
G=gap.SmallGroup(288,301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,254,135,142,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations