metabelian, supersoluble, monomial
Aliases: C24.20D6, C32⋊10SD32, (C3×Q16)⋊1S3, (C3×C6).39D8, Q16⋊1(C3⋊S3), (C3×C12).54D4, C24.S3⋊5C2, C32⋊5D8.3C2, C6.25(D4⋊S3), C3⋊3(C8.6D6), (C32×Q16)⋊2C2, C12.36(C3⋊D4), (C3×C24).19C22, C2.6(C32⋊7D8), C4.3(C32⋊7D4), C8.6(C2×C3⋊S3), SmallGroup(288,303)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊10SD32
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c7 >
Subgroups: 440 in 78 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, D4, Q8, C32, C12, C12, D6, C16, D8, Q16, C3⋊S3, C3×C6, C24, D12, C3×Q8, SD32, C3×C12, C3×C12, C2×C3⋊S3, C3⋊C16, D24, C3×Q16, C3×C24, C12⋊S3, Q8×C32, C8.6D6, C24.S3, C32⋊5D8, C32×Q16, C32⋊10SD32
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, C3⋊D4, SD32, C2×C3⋊S3, D4⋊S3, C32⋊7D4, C8.6D6, C32⋊7D8, C32⋊10SD32
►On 144 points
Generators in S144
(1 59 40)(2 41 60)(3 61 42)(4 43 62)(5 63 44)(6 45 64)(7 49 46)(8 47 50)(9 51 48)(10 33 52)(11 53 34)(12 35 54)(13 55 36)(14 37 56)(15 57 38)(16 39 58)(17 122 86)(18 87 123)(19 124 88)(20 89 125)(21 126 90)(22 91 127)(23 128 92)(24 93 113)(25 114 94)(26 95 115)(27 116 96)(28 81 117)(29 118 82)(30 83 119)(31 120 84)(32 85 121)(65 111 140)(66 141 112)(67 97 142)(68 143 98)(69 99 144)(70 129 100)(71 101 130)(72 131 102)(73 103 132)(74 133 104)(75 105 134)(76 135 106)(77 107 136)(78 137 108)(79 109 138)(80 139 110)
(1 130 126)(2 127 131)(3 132 128)(4 113 133)(5 134 114)(6 115 135)(7 136 116)(8 117 137)(9 138 118)(10 119 139)(11 140 120)(12 121 141)(13 142 122)(14 123 143)(15 144 124)(16 125 129)(17 36 97)(18 98 37)(19 38 99)(20 100 39)(21 40 101)(22 102 41)(23 42 103)(24 104 43)(25 44 105)(26 106 45)(27 46 107)(28 108 47)(29 48 109)(30 110 33)(31 34 111)(32 112 35)(49 77 96)(50 81 78)(51 79 82)(52 83 80)(53 65 84)(54 85 66)(55 67 86)(56 87 68)(57 69 88)(58 89 70)(59 71 90)(60 91 72)(61 73 92)(62 93 74)(63 75 94)(64 95 76)
(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 75)(18 66)(19 73)(20 80)(21 71)(22 78)(23 69)(24 76)(25 67)(26 74)(27 65)(28 72)(29 79)(30 70)(31 77)(32 68)(33 58)(34 49)(35 56)(36 63)(37 54)(38 61)(39 52)(40 59)(41 50)(42 57)(43 64)(44 55)(45 62)(46 53)(47 60)(48 51)(81 102)(82 109)(83 100)(84 107)(85 98)(86 105)(87 112)(88 103)(89 110)(90 101)(91 108)(92 99)(93 106)(94 97)(95 104)(96 111)(113 135)(114 142)(115 133)(116 140)(117 131)(118 138)(119 129)(120 136)(121 143)(122 134)(123 141)(124 132)(125 139)(126 130)(127 137)(128 144)
G:=sub<Sym(144)| (1,59,40)(2,41,60)(3,61,42)(4,43,62)(5,63,44)(6,45,64)(7,49,46)(8,47,50)(9,51,48)(10,33,52)(11,53,34)(12,35,54)(13,55,36)(14,37,56)(15,57,38)(16,39,58)(17,122,86)(18,87,123)(19,124,88)(20,89,125)(21,126,90)(22,91,127)(23,128,92)(24,93,113)(25,114,94)(26,95,115)(27,116,96)(28,81,117)(29,118,82)(30,83,119)(31,120,84)(32,85,121)(65,111,140)(66,141,112)(67,97,142)(68,143,98)(69,99,144)(70,129,100)(71,101,130)(72,131,102)(73,103,132)(74,133,104)(75,105,134)(76,135,106)(77,107,136)(78,137,108)(79,109,138)(80,139,110), (1,130,126)(2,127,131)(3,132,128)(4,113,133)(5,134,114)(6,115,135)(7,136,116)(8,117,137)(9,138,118)(10,119,139)(11,140,120)(12,121,141)(13,142,122)(14,123,143)(15,144,124)(16,125,129)(17,36,97)(18,98,37)(19,38,99)(20,100,39)(21,40,101)(22,102,41)(23,42,103)(24,104,43)(25,44,105)(26,106,45)(27,46,107)(28,108,47)(29,48,109)(30,110,33)(31,34,111)(32,112,35)(49,77,96)(50,81,78)(51,79,82)(52,83,80)(53,65,84)(54,85,66)(55,67,86)(56,87,68)(57,69,88)(58,89,70)(59,71,90)(60,91,72)(61,73,92)(62,93,74)(63,75,94)(64,95,76), (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)(17,75)(18,66)(19,73)(20,80)(21,71)(22,78)(23,69)(24,76)(25,67)(26,74)(27,65)(28,72)(29,79)(30,70)(31,77)(32,68)(33,58)(34,49)(35,56)(36,63)(37,54)(38,61)(39,52)(40,59)(41,50)(42,57)(43,64)(44,55)(45,62)(46,53)(47,60)(48,51)(81,102)(82,109)(83,100)(84,107)(85,98)(86,105)(87,112)(88,103)(89,110)(90,101)(91,108)(92,99)(93,106)(94,97)(95,104)(96,111)(113,135)(114,142)(115,133)(116,140)(117,131)(118,138)(119,129)(120,136)(121,143)(122,134)(123,141)(124,132)(125,139)(126,130)(127,137)(128,144)>;
G:=Group( (1,59,40)(2,41,60)(3,61,42)(4,43,62)(5,63,44)(6,45,64)(7,49,46)(8,47,50)(9,51,48)(10,33,52)(11,53,34)(12,35,54)(13,55,36)(14,37,56)(15,57,38)(16,39,58)(17,122,86)(18,87,123)(19,124,88)(20,89,125)(21,126,90)(22,91,127)(23,128,92)(24,93,113)(25,114,94)(26,95,115)(27,116,96)(28,81,117)(29,118,82)(30,83,119)(31,120,84)(32,85,121)(65,111,140)(66,141,112)(67,97,142)(68,143,98)(69,99,144)(70,129,100)(71,101,130)(72,131,102)(73,103,132)(74,133,104)(75,105,134)(76,135,106)(77,107,136)(78,137,108)(79,109,138)(80,139,110), (1,130,126)(2,127,131)(3,132,128)(4,113,133)(5,134,114)(6,115,135)(7,136,116)(8,117,137)(9,138,118)(10,119,139)(11,140,120)(12,121,141)(13,142,122)(14,123,143)(15,144,124)(16,125,129)(17,36,97)(18,98,37)(19,38,99)(20,100,39)(21,40,101)(22,102,41)(23,42,103)(24,104,43)(25,44,105)(26,106,45)(27,46,107)(28,108,47)(29,48,109)(30,110,33)(31,34,111)(32,112,35)(49,77,96)(50,81,78)(51,79,82)(52,83,80)(53,65,84)(54,85,66)(55,67,86)(56,87,68)(57,69,88)(58,89,70)(59,71,90)(60,91,72)(61,73,92)(62,93,74)(63,75,94)(64,95,76), (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)(17,75)(18,66)(19,73)(20,80)(21,71)(22,78)(23,69)(24,76)(25,67)(26,74)(27,65)(28,72)(29,79)(30,70)(31,77)(32,68)(33,58)(34,49)(35,56)(36,63)(37,54)(38,61)(39,52)(40,59)(41,50)(42,57)(43,64)(44,55)(45,62)(46,53)(47,60)(48,51)(81,102)(82,109)(83,100)(84,107)(85,98)(86,105)(87,112)(88,103)(89,110)(90,101)(91,108)(92,99)(93,106)(94,97)(95,104)(96,111)(113,135)(114,142)(115,133)(116,140)(117,131)(118,138)(119,129)(120,136)(121,143)(122,134)(123,141)(124,132)(125,139)(126,130)(127,137)(128,144) );
G=PermutationGroup([[(1,59,40),(2,41,60),(3,61,42),(4,43,62),(5,63,44),(6,45,64),(7,49,46),(8,47,50),(9,51,48),(10,33,52),(11,53,34),(12,35,54),(13,55,36),(14,37,56),(15,57,38),(16,39,58),(17,122,86),(18,87,123),(19,124,88),(20,89,125),(21,126,90),(22,91,127),(23,128,92),(24,93,113),(25,114,94),(26,95,115),(27,116,96),(28,81,117),(29,118,82),(30,83,119),(31,120,84),(32,85,121),(65,111,140),(66,141,112),(67,97,142),(68,143,98),(69,99,144),(70,129,100),(71,101,130),(72,131,102),(73,103,132),(74,133,104),(75,105,134),(76,135,106),(77,107,136),(78,137,108),(79,109,138),(80,139,110)], [(1,130,126),(2,127,131),(3,132,128),(4,113,133),(5,134,114),(6,115,135),(7,136,116),(8,117,137),(9,138,118),(10,119,139),(11,140,120),(12,121,141),(13,142,122),(14,123,143),(15,144,124),(16,125,129),(17,36,97),(18,98,37),(19,38,99),(20,100,39),(21,40,101),(22,102,41),(23,42,103),(24,104,43),(25,44,105),(26,106,45),(27,46,107),(28,108,47),(29,48,109),(30,110,33),(31,34,111),(32,112,35),(49,77,96),(50,81,78),(51,79,82),(52,83,80),(53,65,84),(54,85,66),(55,67,86),(56,87,68),(57,69,88),(58,89,70),(59,71,90),(60,91,72),(61,73,92),(62,93,74),(63,75,94),(64,95,76)], [(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),(17,75),(18,66),(19,73),(20,80),(21,71),(22,78),(23,69),(24,76),(25,67),(26,74),(27,65),(28,72),(29,79),(30,70),(31,77),(32,68),(33,58),(34,49),(35,56),(36,63),(37,54),(38,61),(39,52),(40,59),(41,50),(42,57),(43,64),(44,55),(45,62),(46,53),(47,60),(48,51),(81,102),(82,109),(83,100),(84,107),(85,98),(86,105),(87,112),(88,103),(89,110),(90,101),(91,108),(92,99),(93,106),(94,97),(95,104),(96,111),(113,135),(114,142),(115,133),(116,140),(117,131),(118,138),(119,129),(120,136),(121,143),(122,134),(123,141),(124,132),(125,139),(126,130),(127,137),(128,144)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 16A | 16B | 16C | 16D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 72 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 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 | SD32 | D4⋊S3 | C8.6D6 |
| kernel | C32⋊10SD32 | C24.S3 | C32⋊5D8 | C32×Q16 | C3×Q16 | 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⋊10SD32 ►in GL6(𝔽97)
| 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 96 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 41 | 56 | 0 | 0 | 0 | 0 |
| 15 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 77 |
| 0 | 0 | 0 | 0 | 10 | 34 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 96 |
G:=sub<GL(6,GF(97))| [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],[0,96,0,0,0,0,1,96,0,0,0,0,0,0,0,96,0,0,0,0,1,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[41,15,0,0,0,0,56,56,0,0,0,0,0,0,96,0,0,0,0,0,96,1,0,0,0,0,0,0,54,10,0,0,0,0,77,34],[1,0,0,0,0,0,1,96,0,0,0,0,0,0,1,0,0,0,0,0,1,96,0,0,0,0,0,0,1,1,0,0,0,0,0,96] >;
C32⋊10SD32 in GAP, Magma, Sage, TeX
C_3^2\rtimes_{10}{\rm SD}_{32} % in TeX
G:=Group("C3^2:10SD32"); // GroupNames label
G:=SmallGroup(288,303);
// by ID
G=gap.SmallGroup(288,303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,120,254,135,142,675,346,80,2693,9414]);
// 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^7>;
// generators/relations