direct product, metabelian, supersoluble, monomial
Aliases: C3×C32⋊7Q16, C33⋊14Q16, C12.35(S3×C6), (C3×C12).130D6, C32⋊4C8.3C6, C32⋊10(C3×Q16), (C32×C6).76D4, (Q8×C33).2C2, C32⋊4Q8.5C6, (Q8×C32).26S3, (Q8×C32).23C6, C32⋊12(C3⋊Q16), C6.38(C32⋊7D4), (C32×C12).30C22, C4.4(C6×C3⋊S3), C3⋊3(C3×C3⋊Q16), Q8.3(C3×C3⋊S3), C12.55(C2×C3⋊S3), (C3×C6).72(C3×D4), C6.41(C3×C3⋊D4), (C3×C12).49(C2×C6), (C3×Q8).33(C3×S3), C2.7(C3×C32⋊7D4), (C3×Q8).16(C3⋊S3), (C3×C32⋊4C8).5C2, (C3×C32⋊4Q8).5C2, (C3×C6).111(C3⋊D4), SmallGroup(432,494)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊7Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 388 in 156 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C32⋊4Q8, Q8×C32, Q8×C32, Q8×C32, C3×C3⋊Dic3, C32×C12, C32×C12, C3×C3⋊Q16, C32⋊7Q16, C3×C32⋊4C8, C3×C32⋊4Q8, Q8×C33, C3×C32⋊7Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊S3, C3⋊D4, C3×D4, S3×C6, C2×C3⋊S3, C3⋊Q16, C3×Q16, C3×C3⋊S3, C3×C3⋊D4, C32⋊7D4, C6×C3⋊S3, C3×C3⋊Q16, C32⋊7Q16, C3×C32⋊7D4, C3×C32⋊7Q16
►On 144 points
Generators in S144
(1 31 106)(2 32 107)(3 25 108)(4 26 109)(5 27 110)(6 28 111)(7 29 112)(8 30 105)(9 95 22)(10 96 23)(11 89 24)(12 90 17)(13 91 18)(14 92 19)(15 93 20)(16 94 21)(33 102 137)(34 103 138)(35 104 139)(36 97 140)(37 98 141)(38 99 142)(39 100 143)(40 101 144)(41 121 71)(42 122 72)(43 123 65)(44 124 66)(45 125 67)(46 126 68)(47 127 69)(48 128 70)(49 120 134)(50 113 135)(51 114 136)(52 115 129)(53 116 130)(54 117 131)(55 118 132)(56 119 133)(57 79 82)(58 80 83)(59 73 84)(60 74 85)(61 75 86)(62 76 87)(63 77 88)(64 78 81)
(1 72 11)(2 12 65)(3 66 13)(4 14 67)(5 68 15)(6 16 69)(7 70 9)(8 10 71)(17 123 107)(18 108 124)(19 125 109)(20 110 126)(21 127 111)(22 112 128)(23 121 105)(24 106 122)(25 44 91)(26 92 45)(27 46 93)(28 94 47)(29 48 95)(30 96 41)(31 42 89)(32 90 43)(33 113 79)(34 80 114)(35 115 73)(36 74 116)(37 117 75)(38 76 118)(39 119 77)(40 78 120)(49 144 64)(50 57 137)(51 138 58)(52 59 139)(53 140 60)(54 61 141)(55 142 62)(56 63 143)(81 134 101)(82 102 135)(83 136 103)(84 104 129)(85 130 97)(86 98 131)(87 132 99)(88 100 133)
(1 106 31)(2 32 107)(3 108 25)(4 26 109)(5 110 27)(6 28 111)(7 112 29)(8 30 105)(9 22 95)(10 96 23)(11 24 89)(12 90 17)(13 18 91)(14 92 19)(15 20 93)(16 94 21)(33 137 102)(34 103 138)(35 139 104)(36 97 140)(37 141 98)(38 99 142)(39 143 100)(40 101 144)(41 121 71)(42 72 122)(43 123 65)(44 66 124)(45 125 67)(46 68 126)(47 127 69)(48 70 128)(49 120 134)(50 135 113)(51 114 136)(52 129 115)(53 116 130)(54 131 117)(55 118 132)(56 133 119)(57 82 79)(58 80 83)(59 84 73)(60 74 85)(61 86 75)(62 76 87)(63 88 77)(64 78 81)
(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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 56 13 52)(10 55 14 51)(11 54 15 50)(12 53 16 49)(17 130 21 134)(18 129 22 133)(19 136 23 132)(20 135 24 131)(25 73 29 77)(26 80 30 76)(27 79 31 75)(28 78 32 74)(33 42 37 46)(34 41 38 45)(35 48 39 44)(36 47 40 43)(65 140 69 144)(66 139 70 143)(67 138 71 142)(68 137 72 141)(81 107 85 111)(82 106 86 110)(83 105 87 109)(84 112 88 108)(89 117 93 113)(90 116 94 120)(91 115 95 119)(92 114 96 118)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)
G:=sub<Sym(144)| (1,31,106)(2,32,107)(3,25,108)(4,26,109)(5,27,110)(6,28,111)(7,29,112)(8,30,105)(9,95,22)(10,96,23)(11,89,24)(12,90,17)(13,91,18)(14,92,19)(15,93,20)(16,94,21)(33,102,137)(34,103,138)(35,104,139)(36,97,140)(37,98,141)(38,99,142)(39,100,143)(40,101,144)(41,121,71)(42,122,72)(43,123,65)(44,124,66)(45,125,67)(46,126,68)(47,127,69)(48,128,70)(49,120,134)(50,113,135)(51,114,136)(52,115,129)(53,116,130)(54,117,131)(55,118,132)(56,119,133)(57,79,82)(58,80,83)(59,73,84)(60,74,85)(61,75,86)(62,76,87)(63,77,88)(64,78,81), (1,72,11)(2,12,65)(3,66,13)(4,14,67)(5,68,15)(6,16,69)(7,70,9)(8,10,71)(17,123,107)(18,108,124)(19,125,109)(20,110,126)(21,127,111)(22,112,128)(23,121,105)(24,106,122)(25,44,91)(26,92,45)(27,46,93)(28,94,47)(29,48,95)(30,96,41)(31,42,89)(32,90,43)(33,113,79)(34,80,114)(35,115,73)(36,74,116)(37,117,75)(38,76,118)(39,119,77)(40,78,120)(49,144,64)(50,57,137)(51,138,58)(52,59,139)(53,140,60)(54,61,141)(55,142,62)(56,63,143)(81,134,101)(82,102,135)(83,136,103)(84,104,129)(85,130,97)(86,98,131)(87,132,99)(88,100,133), (1,106,31)(2,32,107)(3,108,25)(4,26,109)(5,110,27)(6,28,111)(7,112,29)(8,30,105)(9,22,95)(10,96,23)(11,24,89)(12,90,17)(13,18,91)(14,92,19)(15,20,93)(16,94,21)(33,137,102)(34,103,138)(35,139,104)(36,97,140)(37,141,98)(38,99,142)(39,143,100)(40,101,144)(41,121,71)(42,72,122)(43,123,65)(44,66,124)(45,125,67)(46,68,126)(47,127,69)(48,70,128)(49,120,134)(50,135,113)(51,114,136)(52,129,115)(53,116,130)(54,131,117)(55,118,132)(56,133,119)(57,82,79)(58,80,83)(59,84,73)(60,74,85)(61,86,75)(62,76,87)(63,88,77)(64,78,81), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,56,13,52)(10,55,14,51)(11,54,15,50)(12,53,16,49)(17,130,21,134)(18,129,22,133)(19,136,23,132)(20,135,24,131)(25,73,29,77)(26,80,30,76)(27,79,31,75)(28,78,32,74)(33,42,37,46)(34,41,38,45)(35,48,39,44)(36,47,40,43)(65,140,69,144)(66,139,70,143)(67,138,71,142)(68,137,72,141)(81,107,85,111)(82,106,86,110)(83,105,87,109)(84,112,88,108)(89,117,93,113)(90,116,94,120)(91,115,95,119)(92,114,96,118)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128)>;
G:=Group( (1,31,106)(2,32,107)(3,25,108)(4,26,109)(5,27,110)(6,28,111)(7,29,112)(8,30,105)(9,95,22)(10,96,23)(11,89,24)(12,90,17)(13,91,18)(14,92,19)(15,93,20)(16,94,21)(33,102,137)(34,103,138)(35,104,139)(36,97,140)(37,98,141)(38,99,142)(39,100,143)(40,101,144)(41,121,71)(42,122,72)(43,123,65)(44,124,66)(45,125,67)(46,126,68)(47,127,69)(48,128,70)(49,120,134)(50,113,135)(51,114,136)(52,115,129)(53,116,130)(54,117,131)(55,118,132)(56,119,133)(57,79,82)(58,80,83)(59,73,84)(60,74,85)(61,75,86)(62,76,87)(63,77,88)(64,78,81), (1,72,11)(2,12,65)(3,66,13)(4,14,67)(5,68,15)(6,16,69)(7,70,9)(8,10,71)(17,123,107)(18,108,124)(19,125,109)(20,110,126)(21,127,111)(22,112,128)(23,121,105)(24,106,122)(25,44,91)(26,92,45)(27,46,93)(28,94,47)(29,48,95)(30,96,41)(31,42,89)(32,90,43)(33,113,79)(34,80,114)(35,115,73)(36,74,116)(37,117,75)(38,76,118)(39,119,77)(40,78,120)(49,144,64)(50,57,137)(51,138,58)(52,59,139)(53,140,60)(54,61,141)(55,142,62)(56,63,143)(81,134,101)(82,102,135)(83,136,103)(84,104,129)(85,130,97)(86,98,131)(87,132,99)(88,100,133), (1,106,31)(2,32,107)(3,108,25)(4,26,109)(5,110,27)(6,28,111)(7,112,29)(8,30,105)(9,22,95)(10,96,23)(11,24,89)(12,90,17)(13,18,91)(14,92,19)(15,20,93)(16,94,21)(33,137,102)(34,103,138)(35,139,104)(36,97,140)(37,141,98)(38,99,142)(39,143,100)(40,101,144)(41,121,71)(42,72,122)(43,123,65)(44,66,124)(45,125,67)(46,68,126)(47,127,69)(48,70,128)(49,120,134)(50,135,113)(51,114,136)(52,129,115)(53,116,130)(54,131,117)(55,118,132)(56,133,119)(57,82,79)(58,80,83)(59,84,73)(60,74,85)(61,86,75)(62,76,87)(63,88,77)(64,78,81), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,56,13,52)(10,55,14,51)(11,54,15,50)(12,53,16,49)(17,130,21,134)(18,129,22,133)(19,136,23,132)(20,135,24,131)(25,73,29,77)(26,80,30,76)(27,79,31,75)(28,78,32,74)(33,42,37,46)(34,41,38,45)(35,48,39,44)(36,47,40,43)(65,140,69,144)(66,139,70,143)(67,138,71,142)(68,137,72,141)(81,107,85,111)(82,106,86,110)(83,105,87,109)(84,112,88,108)(89,117,93,113)(90,116,94,120)(91,115,95,119)(92,114,96,118)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128) );
G=PermutationGroup([[(1,31,106),(2,32,107),(3,25,108),(4,26,109),(5,27,110),(6,28,111),(7,29,112),(8,30,105),(9,95,22),(10,96,23),(11,89,24),(12,90,17),(13,91,18),(14,92,19),(15,93,20),(16,94,21),(33,102,137),(34,103,138),(35,104,139),(36,97,140),(37,98,141),(38,99,142),(39,100,143),(40,101,144),(41,121,71),(42,122,72),(43,123,65),(44,124,66),(45,125,67),(46,126,68),(47,127,69),(48,128,70),(49,120,134),(50,113,135),(51,114,136),(52,115,129),(53,116,130),(54,117,131),(55,118,132),(56,119,133),(57,79,82),(58,80,83),(59,73,84),(60,74,85),(61,75,86),(62,76,87),(63,77,88),(64,78,81)], [(1,72,11),(2,12,65),(3,66,13),(4,14,67),(5,68,15),(6,16,69),(7,70,9),(8,10,71),(17,123,107),(18,108,124),(19,125,109),(20,110,126),(21,127,111),(22,112,128),(23,121,105),(24,106,122),(25,44,91),(26,92,45),(27,46,93),(28,94,47),(29,48,95),(30,96,41),(31,42,89),(32,90,43),(33,113,79),(34,80,114),(35,115,73),(36,74,116),(37,117,75),(38,76,118),(39,119,77),(40,78,120),(49,144,64),(50,57,137),(51,138,58),(52,59,139),(53,140,60),(54,61,141),(55,142,62),(56,63,143),(81,134,101),(82,102,135),(83,136,103),(84,104,129),(85,130,97),(86,98,131),(87,132,99),(88,100,133)], [(1,106,31),(2,32,107),(3,108,25),(4,26,109),(5,110,27),(6,28,111),(7,112,29),(8,30,105),(9,22,95),(10,96,23),(11,24,89),(12,90,17),(13,18,91),(14,92,19),(15,20,93),(16,94,21),(33,137,102),(34,103,138),(35,139,104),(36,97,140),(37,141,98),(38,99,142),(39,143,100),(40,101,144),(41,121,71),(42,72,122),(43,123,65),(44,66,124),(45,125,67),(46,68,126),(47,127,69),(48,70,128),(49,120,134),(50,135,113),(51,114,136),(52,129,115),(53,116,130),(54,131,117),(55,118,132),(56,133,119),(57,82,79),(58,80,83),(59,84,73),(60,74,85),(61,86,75),(62,76,87),(63,88,77),(64,78,81)], [(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)], [(1,61,5,57),(2,60,6,64),(3,59,7,63),(4,58,8,62),(9,56,13,52),(10,55,14,51),(11,54,15,50),(12,53,16,49),(17,130,21,134),(18,129,22,133),(19,136,23,132),(20,135,24,131),(25,73,29,77),(26,80,30,76),(27,79,31,75),(28,78,32,74),(33,42,37,46),(34,41,38,45),(35,48,39,44),(36,47,40,43),(65,140,69,144),(66,139,70,143),(67,138,71,142),(68,137,72,141),(81,107,85,111),(82,106,86,110),(83,105,87,109),(84,112,88,108),(89,117,93,113),(90,116,94,120),(91,115,95,119),(92,114,96,118),(97,127,101,123),(98,126,102,122),(99,125,103,121),(100,124,104,128)]])
81 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6N | 8A | 8B | 12A | 12B | 12C | ··· | 12AN | 12AO | 12AP | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 4 | 36 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 36 | 36 | 18 | 18 | 18 | 18 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | Q16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×Q16 | C3×C3⋊D4 | C3⋊Q16 | C3×C3⋊Q16 |
| kernel | C3×C32⋊7Q16 | C3×C32⋊4C8 | C3×C32⋊4Q8 | Q8×C33 | C32⋊7Q16 | C32⋊4C8 | C32⋊4Q8 | Q8×C32 | Q8×C32 | C32×C6 | C3×C12 | C33 | C3×Q8 | C3×C6 | C3×C6 | C12 | C32 | C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 2 | 8 | 8 | 2 | 8 | 4 | 16 | 4 | 8 |
Matrix representation of C3×C32⋊7Q16 ►in GL6(𝔽73)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 36 | 64 | 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 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 37 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 63 | 0 | 0 | 0 | 0 |
| 9 | 69 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 59 |
| 0 | 0 | 0 | 0 | 47 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 30 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 0 | 0 | 13 | 56 |
G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,36,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,37,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,9,0,0,0,0,63,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,47,0,0,0,0,59,32],[1,30,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,13,0,0,0,0,17,56] >;
C3×C32⋊7Q16 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_7Q_{16} % in TeX
G:=Group("C3xC3^2:7Q16"); // GroupNames label
G:=SmallGroup(432,494);
// by ID
G=gap.SmallGroup(432,494);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations