direct product, metacyclic, nilpotent (class 4), monomial
Aliases: C32×SD32, C48⋊4C6, C8.3C62, D8.(C3×C6), (C3×C48)⋊6C2, C16⋊2(C3×C6), (C3×Q16)⋊5C6, Q16⋊1(C3×C6), (C3×D8).4C6, (C3×C6).45D8, C6.22(C3×D8), C24.28(C2×C6), C12.46(C3×D4), C2.4(C32×D8), C4.2(D4×C32), (C3×C12).143D4, (C32×Q16)⋊9C2, (C32×D8).3C2, (C3×C24).61C22, SmallGroup(288,330)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
Subgroups: 168 in 78 conjugacy classes, 48 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, C12, C12, C2×C6, C16, D8, Q16, C3×C6, C3×C6, C24, C3×D4, C3×Q8, SD32, C3×C12, C3×C12, C62, C48, C3×D8, C3×Q16, C3×C24, D4×C32, Q8×C32, C3×SD32, C3×C48, C32×D8, C32×Q16, C32×SD32
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, SD32, C62, C3×D8, D4×C32, C3×SD32, C32×D8, C32×SD32
►On 144 points
Generators in S144
(1 54 143)(2 55 144)(3 56 129)(4 57 130)(5 58 131)(6 59 132)(7 60 133)(8 61 134)(9 62 135)(10 63 136)(11 64 137)(12 49 138)(13 50 139)(14 51 140)(15 52 141)(16 53 142)(17 124 37)(18 125 38)(19 126 39)(20 127 40)(21 128 41)(22 113 42)(23 114 43)(24 115 44)(25 116 45)(26 117 46)(27 118 47)(28 119 48)(29 120 33)(30 121 34)(31 122 35)(32 123 36)(65 91 105)(66 92 106)(67 93 107)(68 94 108)(69 95 109)(70 96 110)(71 81 111)(72 82 112)(73 83 97)(74 84 98)(75 85 99)(76 86 100)(77 87 101)(78 88 102)(79 89 103)(80 90 104)
(1 81 30)(2 82 31)(3 83 32)(4 84 17)(5 85 18)(6 86 19)(7 87 20)(8 88 21)(9 89 22)(10 90 23)(11 91 24)(12 92 25)(13 93 26)(14 94 27)(15 95 28)(16 96 29)(33 142 70)(34 143 71)(35 144 72)(36 129 73)(37 130 74)(38 131 75)(39 132 76)(40 133 77)(41 134 78)(42 135 79)(43 136 80)(44 137 65)(45 138 66)(46 139 67)(47 140 68)(48 141 69)(49 106 116)(50 107 117)(51 108 118)(52 109 119)(53 110 120)(54 111 121)(55 112 122)(56 97 123)(57 98 124)(58 99 125)(59 100 126)(60 101 127)(61 102 128)(62 103 113)(63 104 114)(64 105 115)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 19)(18 26)(20 24)(21 31)(23 29)(25 27)(28 32)(33 43)(35 41)(36 48)(37 39)(38 46)(40 44)(45 47)(49 51)(50 58)(52 56)(53 63)(55 61)(57 59)(60 64)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)(82 88)(83 95)(84 86)(85 93)(87 91)(90 96)(92 94)(97 109)(98 100)(99 107)(101 105)(102 112)(104 110)(106 108)(114 120)(115 127)(116 118)(117 125)(119 123)(122 128)(124 126)(129 141)(130 132)(131 139)(133 137)(134 144)(136 142)(138 140)
G:=sub<Sym(144)| (1,54,143)(2,55,144)(3,56,129)(4,57,130)(5,58,131)(6,59,132)(7,60,133)(8,61,134)(9,62,135)(10,63,136)(11,64,137)(12,49,138)(13,50,139)(14,51,140)(15,52,141)(16,53,142)(17,124,37)(18,125,38)(19,126,39)(20,127,40)(21,128,41)(22,113,42)(23,114,43)(24,115,44)(25,116,45)(26,117,46)(27,118,47)(28,119,48)(29,120,33)(30,121,34)(31,122,35)(32,123,36)(65,91,105)(66,92,106)(67,93,107)(68,94,108)(69,95,109)(70,96,110)(71,81,111)(72,82,112)(73,83,97)(74,84,98)(75,85,99)(76,86,100)(77,87,101)(78,88,102)(79,89,103)(80,90,104), (1,81,30)(2,82,31)(3,83,32)(4,84,17)(5,85,18)(6,86,19)(7,87,20)(8,88,21)(9,89,22)(10,90,23)(11,91,24)(12,92,25)(13,93,26)(14,94,27)(15,95,28)(16,96,29)(33,142,70)(34,143,71)(35,144,72)(36,129,73)(37,130,74)(38,131,75)(39,132,76)(40,133,77)(41,134,78)(42,135,79)(43,136,80)(44,137,65)(45,138,66)(46,139,67)(47,140,68)(48,141,69)(49,106,116)(50,107,117)(51,108,118)(52,109,119)(53,110,120)(54,111,121)(55,112,122)(56,97,123)(57,98,124)(58,99,125)(59,100,126)(60,101,127)(61,102,128)(62,103,113)(63,104,114)(64,105,115), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,43)(35,41)(36,48)(37,39)(38,46)(40,44)(45,47)(49,51)(50,58)(52,56)(53,63)(55,61)(57,59)(60,64)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(82,88)(83,95)(84,86)(85,93)(87,91)(90,96)(92,94)(97,109)(98,100)(99,107)(101,105)(102,112)(104,110)(106,108)(114,120)(115,127)(116,118)(117,125)(119,123)(122,128)(124,126)(129,141)(130,132)(131,139)(133,137)(134,144)(136,142)(138,140)>;
G:=Group( (1,54,143)(2,55,144)(3,56,129)(4,57,130)(5,58,131)(6,59,132)(7,60,133)(8,61,134)(9,62,135)(10,63,136)(11,64,137)(12,49,138)(13,50,139)(14,51,140)(15,52,141)(16,53,142)(17,124,37)(18,125,38)(19,126,39)(20,127,40)(21,128,41)(22,113,42)(23,114,43)(24,115,44)(25,116,45)(26,117,46)(27,118,47)(28,119,48)(29,120,33)(30,121,34)(31,122,35)(32,123,36)(65,91,105)(66,92,106)(67,93,107)(68,94,108)(69,95,109)(70,96,110)(71,81,111)(72,82,112)(73,83,97)(74,84,98)(75,85,99)(76,86,100)(77,87,101)(78,88,102)(79,89,103)(80,90,104), (1,81,30)(2,82,31)(3,83,32)(4,84,17)(5,85,18)(6,86,19)(7,87,20)(8,88,21)(9,89,22)(10,90,23)(11,91,24)(12,92,25)(13,93,26)(14,94,27)(15,95,28)(16,96,29)(33,142,70)(34,143,71)(35,144,72)(36,129,73)(37,130,74)(38,131,75)(39,132,76)(40,133,77)(41,134,78)(42,135,79)(43,136,80)(44,137,65)(45,138,66)(46,139,67)(47,140,68)(48,141,69)(49,106,116)(50,107,117)(51,108,118)(52,109,119)(53,110,120)(54,111,121)(55,112,122)(56,97,123)(57,98,124)(58,99,125)(59,100,126)(60,101,127)(61,102,128)(62,103,113)(63,104,114)(64,105,115), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,43)(35,41)(36,48)(37,39)(38,46)(40,44)(45,47)(49,51)(50,58)(52,56)(53,63)(55,61)(57,59)(60,64)(65,77)(66,68)(67,75)(69,73)(70,80)(72,78)(74,76)(82,88)(83,95)(84,86)(85,93)(87,91)(90,96)(92,94)(97,109)(98,100)(99,107)(101,105)(102,112)(104,110)(106,108)(114,120)(115,127)(116,118)(117,125)(119,123)(122,128)(124,126)(129,141)(130,132)(131,139)(133,137)(134,144)(136,142)(138,140) );
G=PermutationGroup([[(1,54,143),(2,55,144),(3,56,129),(4,57,130),(5,58,131),(6,59,132),(7,60,133),(8,61,134),(9,62,135),(10,63,136),(11,64,137),(12,49,138),(13,50,139),(14,51,140),(15,52,141),(16,53,142),(17,124,37),(18,125,38),(19,126,39),(20,127,40),(21,128,41),(22,113,42),(23,114,43),(24,115,44),(25,116,45),(26,117,46),(27,118,47),(28,119,48),(29,120,33),(30,121,34),(31,122,35),(32,123,36),(65,91,105),(66,92,106),(67,93,107),(68,94,108),(69,95,109),(70,96,110),(71,81,111),(72,82,112),(73,83,97),(74,84,98),(75,85,99),(76,86,100),(77,87,101),(78,88,102),(79,89,103),(80,90,104)], [(1,81,30),(2,82,31),(3,83,32),(4,84,17),(5,85,18),(6,86,19),(7,87,20),(8,88,21),(9,89,22),(10,90,23),(11,91,24),(12,92,25),(13,93,26),(14,94,27),(15,95,28),(16,96,29),(33,142,70),(34,143,71),(35,144,72),(36,129,73),(37,130,74),(38,131,75),(39,132,76),(40,133,77),(41,134,78),(42,135,79),(43,136,80),(44,137,65),(45,138,66),(46,139,67),(47,140,68),(48,141,69),(49,106,116),(50,107,117),(51,108,118),(52,109,119),(53,110,120),(54,111,121),(55,112,122),(56,97,123),(57,98,124),(58,99,125),(59,100,126),(60,101,127),(61,102,128),(62,103,113),(63,104,114),(64,105,115)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,19),(18,26),(20,24),(21,31),(23,29),(25,27),(28,32),(33,43),(35,41),(36,48),(37,39),(38,46),(40,44),(45,47),(49,51),(50,58),(52,56),(53,63),(55,61),(57,59),(60,64),(65,77),(66,68),(67,75),(69,73),(70,80),(72,78),(74,76),(82,88),(83,95),(84,86),(85,93),(87,91),(90,96),(92,94),(97,109),(98,100),(99,107),(101,105),(102,112),(104,110),(106,108),(114,120),(115,127),(116,118),(117,125),(119,123),(122,128),(124,126),(129,141),(130,132),(131,139),(133,137),(134,144),(136,142),(138,140)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12P | 16A | 16B | 16C | 16D | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 8 | 1 | ··· | 1 | 2 | 8 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D8 | C3×D4 | SD32 | C3×D8 | C3×SD32 |
| kernel | C32×SD32 | C3×C48 | C32×D8 | C32×Q16 | C3×SD32 | C48 | C3×D8 | C3×Q16 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 1 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C32×SD32 ►in GL3(𝔽97) generated by
| 61 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 35 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 1 | 0 | 0 |
| 0 | 44 | 87 |
| 0 | 10 | 44 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 96 |
G:=sub<GL(3,GF(97))| [61,0,0,0,1,0,0,0,1],[35,0,0,0,61,0,0,0,61],[1,0,0,0,44,10,0,87,44],[96,0,0,0,1,0,0,0,96] >;
C32×SD32 in GAP, Magma, Sage, TeX
C_3^2\times {\rm SD}_{32} % in TeX
G:=Group("C3^2xSD32"); // GroupNames label
G:=SmallGroup(288,330);
// by ID
G=gap.SmallGroup(288,330);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,1008,533,3784,1901,242,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations