metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊3SD64, Q32⋊1D5, C20.7D8, D80.2C2, C16.6D10, C40.11D4, C10.10D16, C80.4C22, C5⋊2C32⋊3C2, (C5×Q32)⋊1C2, C4.3(D4⋊D5), C8.11(C5⋊D4), C2.6(C5⋊D16), SmallGroup(320,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊SD64
G = < a,b,c | a5=b32=c2=1, bab-1=cac=a-1, cbc=b15 >
Smallest permutation representation of C5⋊SD64
►On 160 points
Generators in S160
(1 63 145 87 112)(2 113 88 146 64)(3 33 147 89 114)(4 115 90 148 34)(5 35 149 91 116)(6 117 92 150 36)(7 37 151 93 118)(8 119 94 152 38)(9 39 153 95 120)(10 121 96 154 40)(11 41 155 65 122)(12 123 66 156 42)(13 43 157 67 124)(14 125 68 158 44)(15 45 159 69 126)(16 127 70 160 46)(17 47 129 71 128)(18 97 72 130 48)(19 49 131 73 98)(20 99 74 132 50)(21 51 133 75 100)(22 101 76 134 52)(23 53 135 77 102)(24 103 78 136 54)(25 55 137 79 104)(26 105 80 138 56)(27 57 139 81 106)(28 107 82 140 58)(29 59 141 83 108)(30 109 84 142 60)(31 61 143 85 110)(32 111 86 144 62)
(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)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(2 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 110)(34 125)(35 108)(36 123)(37 106)(38 121)(39 104)(40 119)(41 102)(42 117)(43 100)(44 115)(45 98)(46 113)(47 128)(48 111)(49 126)(50 109)(51 124)(52 107)(53 122)(54 105)(55 120)(56 103)(57 118)(58 101)(59 116)(60 99)(61 114)(62 97)(63 112)(64 127)(65 135)(66 150)(67 133)(68 148)(69 131)(70 146)(71 129)(72 144)(73 159)(74 142)(75 157)(76 140)(77 155)(78 138)(79 153)(80 136)(81 151)(82 134)(83 149)(84 132)(85 147)(86 130)(87 145)(88 160)(89 143)(90 158)(91 141)(92 156)(93 139)(94 154)(95 137)(96 152)
G:=sub<Sym(160)| (1,63,145,87,112)(2,113,88,146,64)(3,33,147,89,114)(4,115,90,148,34)(5,35,149,91,116)(6,117,92,150,36)(7,37,151,93,118)(8,119,94,152,38)(9,39,153,95,120)(10,121,96,154,40)(11,41,155,65,122)(12,123,66,156,42)(13,43,157,67,124)(14,125,68,158,44)(15,45,159,69,126)(16,127,70,160,46)(17,47,129,71,128)(18,97,72,130,48)(19,49,131,73,98)(20,99,74,132,50)(21,51,133,75,100)(22,101,76,134,52)(23,53,135,77,102)(24,103,78,136,54)(25,55,137,79,104)(26,105,80,138,56)(27,57,139,81,106)(28,107,82,140,58)(29,59,141,83,108)(30,109,84,142,60)(31,61,143,85,110)(32,111,86,144,62), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,110)(34,125)(35,108)(36,123)(37,106)(38,121)(39,104)(40,119)(41,102)(42,117)(43,100)(44,115)(45,98)(46,113)(47,128)(48,111)(49,126)(50,109)(51,124)(52,107)(53,122)(54,105)(55,120)(56,103)(57,118)(58,101)(59,116)(60,99)(61,114)(62,97)(63,112)(64,127)(65,135)(66,150)(67,133)(68,148)(69,131)(70,146)(71,129)(72,144)(73,159)(74,142)(75,157)(76,140)(77,155)(78,138)(79,153)(80,136)(81,151)(82,134)(83,149)(84,132)(85,147)(86,130)(87,145)(88,160)(89,143)(90,158)(91,141)(92,156)(93,139)(94,154)(95,137)(96,152)>;
G:=Group( (1,63,145,87,112)(2,113,88,146,64)(3,33,147,89,114)(4,115,90,148,34)(5,35,149,91,116)(6,117,92,150,36)(7,37,151,93,118)(8,119,94,152,38)(9,39,153,95,120)(10,121,96,154,40)(11,41,155,65,122)(12,123,66,156,42)(13,43,157,67,124)(14,125,68,158,44)(15,45,159,69,126)(16,127,70,160,46)(17,47,129,71,128)(18,97,72,130,48)(19,49,131,73,98)(20,99,74,132,50)(21,51,133,75,100)(22,101,76,134,52)(23,53,135,77,102)(24,103,78,136,54)(25,55,137,79,104)(26,105,80,138,56)(27,57,139,81,106)(28,107,82,140,58)(29,59,141,83,108)(30,109,84,142,60)(31,61,143,85,110)(32,111,86,144,62), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,110)(34,125)(35,108)(36,123)(37,106)(38,121)(39,104)(40,119)(41,102)(42,117)(43,100)(44,115)(45,98)(46,113)(47,128)(48,111)(49,126)(50,109)(51,124)(52,107)(53,122)(54,105)(55,120)(56,103)(57,118)(58,101)(59,116)(60,99)(61,114)(62,97)(63,112)(64,127)(65,135)(66,150)(67,133)(68,148)(69,131)(70,146)(71,129)(72,144)(73,159)(74,142)(75,157)(76,140)(77,155)(78,138)(79,153)(80,136)(81,151)(82,134)(83,149)(84,132)(85,147)(86,130)(87,145)(88,160)(89,143)(90,158)(91,141)(92,156)(93,139)(94,154)(95,137)(96,152) );
G=PermutationGroup([[(1,63,145,87,112),(2,113,88,146,64),(3,33,147,89,114),(4,115,90,148,34),(5,35,149,91,116),(6,117,92,150,36),(7,37,151,93,118),(8,119,94,152,38),(9,39,153,95,120),(10,121,96,154,40),(11,41,155,65,122),(12,123,66,156,42),(13,43,157,67,124),(14,125,68,158,44),(15,45,159,69,126),(16,127,70,160,46),(17,47,129,71,128),(18,97,72,130,48),(19,49,131,73,98),(20,99,74,132,50),(21,51,133,75,100),(22,101,76,134,52),(23,53,135,77,102),(24,103,78,136,54),(25,55,137,79,104),(26,105,80,138,56),(27,57,139,81,106),(28,107,82,140,58),(29,59,141,83,108),(30,109,84,142,60),(31,61,143,85,110),(32,111,86,144,62)], [(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),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26),(33,110),(34,125),(35,108),(36,123),(37,106),(38,121),(39,104),(40,119),(41,102),(42,117),(43,100),(44,115),(45,98),(46,113),(47,128),(48,111),(49,126),(50,109),(51,124),(52,107),(53,122),(54,105),(55,120),(56,103),(57,118),(58,101),(59,116),(60,99),(61,114),(62,97),(63,112),(64,127),(65,135),(66,150),(67,133),(68,148),(69,131),(70,146),(71,129),(72,144),(73,159),(74,142),(75,157),(76,140),(77,155),(78,138),(79,153),(80,136),(81,151),(82,134),(83,149),(84,132),(85,147),(86,130),(87,145),(88,160),(89,143),(90,158),(91,141),(92,156),(93,139),(94,154),(95,137),(96,152)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 8A | 8B | 10A | 10B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 32A | ··· | 32H | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 80 | 2 | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 16 | 16 | 16 | 16 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D8 | D10 | D16 | C5⋊D4 | SD64 | D4⋊D5 | C5⋊D16 | C5⋊SD64 |
| kernel | C5⋊SD64 | C5⋊2C32 | D80 | C5×Q32 | C40 | Q32 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C5⋊SD64 ►in GL4(𝔽641) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 640 | 362 |
| 37 | 312 | 0 | 0 |
| 329 | 37 | 0 | 0 |
| 0 | 0 | 419 | 17 |
| 0 | 0 | 419 | 222 |
| 1 | 0 | 0 | 0 |
| 0 | 640 | 0 | 0 |
| 0 | 0 | 362 | 640 |
| 0 | 0 | 279 | 279 |
G:=sub<GL(4,GF(641))| [1,0,0,0,0,1,0,0,0,0,0,640,0,0,1,362],[37,329,0,0,312,37,0,0,0,0,419,419,0,0,17,222],[1,0,0,0,0,640,0,0,0,0,362,279,0,0,640,279] >;
C5⋊SD64 in GAP, Magma, Sage, TeX
C_5\rtimes {\rm SD}_{64} % in TeX
G:=Group("C5:SD64"); // GroupNames label
G:=SmallGroup(320,79);
// by ID
G=gap.SmallGroup(320,79);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,232,254,135,142,675,346,192,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^5=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^15>;
// generators/relations
Export