direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊11SD16, C62.133D4, (C6×Q8)⋊5S3, (C3×Q8)⋊16D6, (C3×C6)⋊11SD16, C6⋊3(Q8⋊2S3), (C3×C12).101D4, (C2×C12).156D6, C32⋊20(C2×SD16), C12.60(C3⋊D4), C4.8(C32⋊7D4), (C6×C12).147C22, (C3×C12).104C23, C12.100(C22×S3), C32⋊4C8⋊24C22, (Q8×C32)⋊15C22, C12⋊S3.29C22, C22.23(C32⋊7D4), (Q8×C3×C6)⋊5C2, Q8⋊4(C2×C3⋊S3), (C2×Q8)⋊3(C3⋊S3), C3⋊4(C2×Q8⋊2S3), (C3×C6).287(C2×D4), C6.128(C2×C3⋊D4), C4.14(C22×C3⋊S3), (C2×C32⋊4C8)⋊11C2, (C2×C12⋊S3).15C2, C2.17(C2×C32⋊7D4), (C2×C6).101(C3⋊D4), (C2×C4).52(C2×C3⋊S3), SmallGroup(288,798)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊11SD16
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d3 >
Subgroups: 884 in 204 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C32, C12, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3×C6, C3⋊C8, D12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C2×SD16, C3×C12, C3×C12, C2×C3⋊S3, C62, C2×C3⋊C8, Q8⋊2S3, C2×D12, C6×Q8, C32⋊4C8, C12⋊S3, C12⋊S3, C6×C12, C6×C12, Q8×C32, Q8×C32, C22×C3⋊S3, C2×Q8⋊2S3, C2×C32⋊4C8, C32⋊11SD16, C2×C12⋊S3, Q8×C3×C6, C2×C32⋊11SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C2×SD16, C2×C3⋊S3, Q8⋊2S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C2×Q8⋊2S3, C32⋊11SD16, C2×C32⋊7D4, C2×C32⋊11SD16
►On 144 points
Generators in S144
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 70)(18 71)(19 72)(20 65)(21 66)(22 67)(23 68)(24 69)(25 144)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)(57 103)(58 104)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(81 94)(82 95)(83 96)(84 89)(85 90)(86 91)(87 92)(88 93)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(111 113)(112 114)(121 135)(122 136)(123 129)(124 130)(125 131)(126 132)(127 133)(128 134)
(1 29 64)(2 57 30)(3 31 58)(4 59 32)(5 25 60)(6 61 26)(7 27 62)(8 63 28)(9 52 129)(10 130 53)(11 54 131)(12 132 55)(13 56 133)(14 134 49)(15 50 135)(16 136 51)(17 120 92)(18 93 113)(19 114 94)(20 95 115)(21 116 96)(22 89 117)(23 118 90)(24 91 119)(33 121 74)(34 75 122)(35 123 76)(36 77 124)(37 125 78)(38 79 126)(39 127 80)(40 73 128)(41 99 137)(42 138 100)(43 101 139)(44 140 102)(45 103 141)(46 142 104)(47 97 143)(48 144 98)(65 82 105)(66 106 83)(67 84 107)(68 108 85)(69 86 109)(70 110 87)(71 88 111)(72 112 81)
(1 22 15)(2 16 23)(3 24 9)(4 10 17)(5 18 11)(6 12 19)(7 20 13)(8 14 21)(25 93 54)(26 55 94)(27 95 56)(28 49 96)(29 89 50)(30 51 90)(31 91 52)(32 53 92)(33 140 84)(34 85 141)(35 142 86)(36 87 143)(37 144 88)(38 81 137)(39 138 82)(40 83 139)(41 79 72)(42 65 80)(43 73 66)(44 67 74)(45 75 68)(46 69 76)(47 77 70)(48 71 78)(57 136 118)(58 119 129)(59 130 120)(60 113 131)(61 132 114)(62 115 133)(63 134 116)(64 117 135)(97 124 110)(98 111 125)(99 126 112)(100 105 127)(101 128 106)(102 107 121)(103 122 108)(104 109 123)
(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 4)(3 7)(6 8)(9 20)(10 23)(11 18)(12 21)(13 24)(14 19)(15 22)(16 17)(25 60)(26 63)(27 58)(28 61)(29 64)(30 59)(31 62)(32 57)(33 107)(34 110)(35 105)(36 108)(37 111)(38 106)(39 109)(40 112)(41 43)(42 46)(45 47)(49 114)(50 117)(51 120)(52 115)(53 118)(54 113)(55 116)(56 119)(65 76)(66 79)(67 74)(68 77)(69 80)(70 75)(71 78)(72 73)(81 128)(82 123)(83 126)(84 121)(85 124)(86 127)(87 122)(88 125)(89 135)(90 130)(91 133)(92 136)(93 131)(94 134)(95 129)(96 132)(97 141)(98 144)(99 139)(100 142)(101 137)(102 140)(103 143)(104 138)
G:=sub<Sym(144)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,76)(10,77)(11,78)(12,79)(13,80)(14,73)(15,74)(16,75)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,144)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49)(57,103)(58,104)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114)(121,135)(122,136)(123,129)(124,130)(125,131)(126,132)(127,133)(128,134), (1,29,64)(2,57,30)(3,31,58)(4,59,32)(5,25,60)(6,61,26)(7,27,62)(8,63,28)(9,52,129)(10,130,53)(11,54,131)(12,132,55)(13,56,133)(14,134,49)(15,50,135)(16,136,51)(17,120,92)(18,93,113)(19,114,94)(20,95,115)(21,116,96)(22,89,117)(23,118,90)(24,91,119)(33,121,74)(34,75,122)(35,123,76)(36,77,124)(37,125,78)(38,79,126)(39,127,80)(40,73,128)(41,99,137)(42,138,100)(43,101,139)(44,140,102)(45,103,141)(46,142,104)(47,97,143)(48,144,98)(65,82,105)(66,106,83)(67,84,107)(68,108,85)(69,86,109)(70,110,87)(71,88,111)(72,112,81), (1,22,15)(2,16,23)(3,24,9)(4,10,17)(5,18,11)(6,12,19)(7,20,13)(8,14,21)(25,93,54)(26,55,94)(27,95,56)(28,49,96)(29,89,50)(30,51,90)(31,91,52)(32,53,92)(33,140,84)(34,85,141)(35,142,86)(36,87,143)(37,144,88)(38,81,137)(39,138,82)(40,83,139)(41,79,72)(42,65,80)(43,73,66)(44,67,74)(45,75,68)(46,69,76)(47,77,70)(48,71,78)(57,136,118)(58,119,129)(59,130,120)(60,113,131)(61,132,114)(62,115,133)(63,134,116)(64,117,135)(97,124,110)(98,111,125)(99,126,112)(100,105,127)(101,128,106)(102,107,121)(103,122,108)(104,109,123), (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), (2,4)(3,7)(6,8)(9,20)(10,23)(11,18)(12,21)(13,24)(14,19)(15,22)(16,17)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57)(33,107)(34,110)(35,105)(36,108)(37,111)(38,106)(39,109)(40,112)(41,43)(42,46)(45,47)(49,114)(50,117)(51,120)(52,115)(53,118)(54,113)(55,116)(56,119)(65,76)(66,79)(67,74)(68,77)(69,80)(70,75)(71,78)(72,73)(81,128)(82,123)(83,126)(84,121)(85,124)(86,127)(87,122)(88,125)(89,135)(90,130)(91,133)(92,136)(93,131)(94,134)(95,129)(96,132)(97,141)(98,144)(99,139)(100,142)(101,137)(102,140)(103,143)(104,138)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,76)(10,77)(11,78)(12,79)(13,80)(14,73)(15,74)(16,75)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,144)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49)(57,103)(58,104)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(81,94)(82,95)(83,96)(84,89)(85,90)(86,91)(87,92)(88,93)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(111,113)(112,114)(121,135)(122,136)(123,129)(124,130)(125,131)(126,132)(127,133)(128,134), (1,29,64)(2,57,30)(3,31,58)(4,59,32)(5,25,60)(6,61,26)(7,27,62)(8,63,28)(9,52,129)(10,130,53)(11,54,131)(12,132,55)(13,56,133)(14,134,49)(15,50,135)(16,136,51)(17,120,92)(18,93,113)(19,114,94)(20,95,115)(21,116,96)(22,89,117)(23,118,90)(24,91,119)(33,121,74)(34,75,122)(35,123,76)(36,77,124)(37,125,78)(38,79,126)(39,127,80)(40,73,128)(41,99,137)(42,138,100)(43,101,139)(44,140,102)(45,103,141)(46,142,104)(47,97,143)(48,144,98)(65,82,105)(66,106,83)(67,84,107)(68,108,85)(69,86,109)(70,110,87)(71,88,111)(72,112,81), (1,22,15)(2,16,23)(3,24,9)(4,10,17)(5,18,11)(6,12,19)(7,20,13)(8,14,21)(25,93,54)(26,55,94)(27,95,56)(28,49,96)(29,89,50)(30,51,90)(31,91,52)(32,53,92)(33,140,84)(34,85,141)(35,142,86)(36,87,143)(37,144,88)(38,81,137)(39,138,82)(40,83,139)(41,79,72)(42,65,80)(43,73,66)(44,67,74)(45,75,68)(46,69,76)(47,77,70)(48,71,78)(57,136,118)(58,119,129)(59,130,120)(60,113,131)(61,132,114)(62,115,133)(63,134,116)(64,117,135)(97,124,110)(98,111,125)(99,126,112)(100,105,127)(101,128,106)(102,107,121)(103,122,108)(104,109,123), (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), (2,4)(3,7)(6,8)(9,20)(10,23)(11,18)(12,21)(13,24)(14,19)(15,22)(16,17)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57)(33,107)(34,110)(35,105)(36,108)(37,111)(38,106)(39,109)(40,112)(41,43)(42,46)(45,47)(49,114)(50,117)(51,120)(52,115)(53,118)(54,113)(55,116)(56,119)(65,76)(66,79)(67,74)(68,77)(69,80)(70,75)(71,78)(72,73)(81,128)(82,123)(83,126)(84,121)(85,124)(86,127)(87,122)(88,125)(89,135)(90,130)(91,133)(92,136)(93,131)(94,134)(95,129)(96,132)(97,141)(98,144)(99,139)(100,142)(101,137)(102,140)(103,143)(104,138) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,76),(10,77),(11,78),(12,79),(13,80),(14,73),(15,74),(16,75),(17,70),(18,71),(19,72),(20,65),(21,66),(22,67),(23,68),(24,69),(25,144),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49),(57,103),(58,104),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(81,94),(82,95),(83,96),(84,89),(85,90),(86,91),(87,92),(88,93),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(111,113),(112,114),(121,135),(122,136),(123,129),(124,130),(125,131),(126,132),(127,133),(128,134)], [(1,29,64),(2,57,30),(3,31,58),(4,59,32),(5,25,60),(6,61,26),(7,27,62),(8,63,28),(9,52,129),(10,130,53),(11,54,131),(12,132,55),(13,56,133),(14,134,49),(15,50,135),(16,136,51),(17,120,92),(18,93,113),(19,114,94),(20,95,115),(21,116,96),(22,89,117),(23,118,90),(24,91,119),(33,121,74),(34,75,122),(35,123,76),(36,77,124),(37,125,78),(38,79,126),(39,127,80),(40,73,128),(41,99,137),(42,138,100),(43,101,139),(44,140,102),(45,103,141),(46,142,104),(47,97,143),(48,144,98),(65,82,105),(66,106,83),(67,84,107),(68,108,85),(69,86,109),(70,110,87),(71,88,111),(72,112,81)], [(1,22,15),(2,16,23),(3,24,9),(4,10,17),(5,18,11),(6,12,19),(7,20,13),(8,14,21),(25,93,54),(26,55,94),(27,95,56),(28,49,96),(29,89,50),(30,51,90),(31,91,52),(32,53,92),(33,140,84),(34,85,141),(35,142,86),(36,87,143),(37,144,88),(38,81,137),(39,138,82),(40,83,139),(41,79,72),(42,65,80),(43,73,66),(44,67,74),(45,75,68),(46,69,76),(47,77,70),(48,71,78),(57,136,118),(58,119,129),(59,130,120),(60,113,131),(61,132,114),(62,115,133),(63,134,116),(64,117,135),(97,124,110),(98,111,125),(99,126,112),(100,105,127),(101,128,106),(102,107,121),(103,122,108),(104,109,123)], [(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)], [(2,4),(3,7),(6,8),(9,20),(10,23),(11,18),(12,21),(13,24),(14,19),(15,22),(16,17),(25,60),(26,63),(27,58),(28,61),(29,64),(30,59),(31,62),(32,57),(33,107),(34,110),(35,105),(36,108),(37,111),(38,106),(39,109),(40,112),(41,43),(42,46),(45,47),(49,114),(50,117),(51,120),(52,115),(53,118),(54,113),(55,116),(56,119),(65,76),(66,79),(67,74),(68,77),(69,80),(70,75),(71,78),(72,73),(81,128),(82,123),(83,126),(84,121),(85,124),(86,127),(87,122),(88,125),(89,135),(90,130),(91,133),(92,136),(93,131),(94,134),(95,129),(96,132),(97,141),(98,144),(99,139),(100,142),(101,137),(102,140),(103,143),(104,138)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | Q8⋊2S3 |
| kernel | C2×C32⋊11SD16 | C2×C32⋊4C8 | C32⋊11SD16 | C2×C12⋊S3 | Q8×C3×C6 | C6×Q8 | C3×C12 | C62 | C2×C12 | C3×Q8 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 4 | 8 | 8 | 8 |
Matrix representation of C2×C32⋊11SD16 ►in GL6(𝔽73)
| 72 | 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 67 | 67 |
| 0 | 0 | 0 | 0 | 6 | 67 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,67,6,0,0,0,0,67,67],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C2×C32⋊11SD16 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_{11}{\rm SD}_{16} % in TeX
G:=Group("C2xC3^2:11SD16"); // GroupNames label
G:=SmallGroup(288,798);
// by ID
G=gap.SmallGroup(288,798);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,100,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
// generators/relations