direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C5⋊Q16, C10⋊2Q16, C20.20D4, Q8.7D10, C20.16C23, Dic10.10C22, C5⋊3(C2×Q16), (C2×Q8).3D5, (C2×C4).54D10, (C2×C10).43D4, C10.55(C2×D4), C4.9(C5⋊D4), (Q8×C10).3C2, C5⋊2C8.9C22, C4.16(C22×D5), (C5×Q8).7C22, (C2×C20).38C22, (C2×Dic10).9C2, C22.24(C5⋊D4), (C2×C5⋊2C8).6C2, C2.19(C2×C5⋊D4), SmallGroup(160,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C5⋊Q16
G = < a,b,c,d | a2=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 152 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×Q16, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C5⋊2C8, C5⋊Q16, C2×Dic10, Q8×C10, C2×C5⋊Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C2×Q16, C5⋊D4, C22×D5, C5⋊Q16, C2×C5⋊D4, C2×C5⋊Q16
►Regular action on 160 points
Generators in S160
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 151)(10 152)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(41 120)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 100)(50 101)(51 102)(52 103)(53 104)(54 97)(55 98)(56 99)(57 108)(58 109)(59 110)(60 111)(61 112)(62 105)(63 106)(64 107)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 153)(144 154)
(1 49 39 146 63)(2 64 147 40 50)(3 51 33 148 57)(4 58 149 34 52)(5 53 35 150 59)(6 60 151 36 54)(7 55 37 152 61)(8 62 145 38 56)(9 21 97 75 111)(10 112 76 98 22)(11 23 99 77 105)(12 106 78 100 24)(13 17 101 79 107)(14 108 80 102 18)(15 19 103 73 109)(16 110 74 104 20)(25 143 86 94 48)(26 41 95 87 144)(27 137 88 96 42)(28 43 89 81 138)(29 139 82 90 44)(30 45 91 83 140)(31 141 84 92 46)(32 47 93 85 142)(65 153 126 134 119)(66 120 135 127 154)(67 155 128 136 113)(68 114 129 121 156)(69 157 122 130 115)(70 116 131 123 158)(71 159 124 132 117)(72 118 133 125 160)
(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)
(1 115 5 119)(2 114 6 118)(3 113 7 117)(4 120 8 116)(9 85 13 81)(10 84 14 88)(11 83 15 87)(12 82 16 86)(17 138 21 142)(18 137 22 141)(19 144 23 140)(20 143 24 139)(25 100 29 104)(26 99 30 103)(27 98 31 102)(28 97 32 101)(33 155 37 159)(34 154 38 158)(35 153 39 157)(36 160 40 156)(41 77 45 73)(42 76 46 80)(43 75 47 79)(44 74 48 78)(49 69 53 65)(50 68 54 72)(51 67 55 71)(52 66 56 70)(57 136 61 132)(58 135 62 131)(59 134 63 130)(60 133 64 129)(89 111 93 107)(90 110 94 106)(91 109 95 105)(92 108 96 112)(121 151 125 147)(122 150 126 146)(123 149 127 145)(124 148 128 152)
G:=sub<Sym(160)| (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,151)(10,152)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(41,120)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,100)(50,101)(51,102)(52,103)(53,104)(54,97)(55,98)(56,99)(57,108)(58,109)(59,110)(60,111)(61,112)(62,105)(63,106)(64,107)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,153)(144,154), (1,49,39,146,63)(2,64,147,40,50)(3,51,33,148,57)(4,58,149,34,52)(5,53,35,150,59)(6,60,151,36,54)(7,55,37,152,61)(8,62,145,38,56)(9,21,97,75,111)(10,112,76,98,22)(11,23,99,77,105)(12,106,78,100,24)(13,17,101,79,107)(14,108,80,102,18)(15,19,103,73,109)(16,110,74,104,20)(25,143,86,94,48)(26,41,95,87,144)(27,137,88,96,42)(28,43,89,81,138)(29,139,82,90,44)(30,45,91,83,140)(31,141,84,92,46)(32,47,93,85,142)(65,153,126,134,119)(66,120,135,127,154)(67,155,128,136,113)(68,114,129,121,156)(69,157,122,130,115)(70,116,131,123,158)(71,159,124,132,117)(72,118,133,125,160), (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), (1,115,5,119)(2,114,6,118)(3,113,7,117)(4,120,8,116)(9,85,13,81)(10,84,14,88)(11,83,15,87)(12,82,16,86)(17,138,21,142)(18,137,22,141)(19,144,23,140)(20,143,24,139)(25,100,29,104)(26,99,30,103)(27,98,31,102)(28,97,32,101)(33,155,37,159)(34,154,38,158)(35,153,39,157)(36,160,40,156)(41,77,45,73)(42,76,46,80)(43,75,47,79)(44,74,48,78)(49,69,53,65)(50,68,54,72)(51,67,55,71)(52,66,56,70)(57,136,61,132)(58,135,62,131)(59,134,63,130)(60,133,64,129)(89,111,93,107)(90,110,94,106)(91,109,95,105)(92,108,96,112)(121,151,125,147)(122,150,126,146)(123,149,127,145)(124,148,128,152)>;
G:=Group( (1,78)(2,79)(3,80)(4,73)(5,74)(6,75)(7,76)(8,77)(9,151)(10,152)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(41,120)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,100)(50,101)(51,102)(52,103)(53,104)(54,97)(55,98)(56,99)(57,108)(58,109)(59,110)(60,111)(61,112)(62,105)(63,106)(64,107)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,153)(144,154), (1,49,39,146,63)(2,64,147,40,50)(3,51,33,148,57)(4,58,149,34,52)(5,53,35,150,59)(6,60,151,36,54)(7,55,37,152,61)(8,62,145,38,56)(9,21,97,75,111)(10,112,76,98,22)(11,23,99,77,105)(12,106,78,100,24)(13,17,101,79,107)(14,108,80,102,18)(15,19,103,73,109)(16,110,74,104,20)(25,143,86,94,48)(26,41,95,87,144)(27,137,88,96,42)(28,43,89,81,138)(29,139,82,90,44)(30,45,91,83,140)(31,141,84,92,46)(32,47,93,85,142)(65,153,126,134,119)(66,120,135,127,154)(67,155,128,136,113)(68,114,129,121,156)(69,157,122,130,115)(70,116,131,123,158)(71,159,124,132,117)(72,118,133,125,160), (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), (1,115,5,119)(2,114,6,118)(3,113,7,117)(4,120,8,116)(9,85,13,81)(10,84,14,88)(11,83,15,87)(12,82,16,86)(17,138,21,142)(18,137,22,141)(19,144,23,140)(20,143,24,139)(25,100,29,104)(26,99,30,103)(27,98,31,102)(28,97,32,101)(33,155,37,159)(34,154,38,158)(35,153,39,157)(36,160,40,156)(41,77,45,73)(42,76,46,80)(43,75,47,79)(44,74,48,78)(49,69,53,65)(50,68,54,72)(51,67,55,71)(52,66,56,70)(57,136,61,132)(58,135,62,131)(59,134,63,130)(60,133,64,129)(89,111,93,107)(90,110,94,106)(91,109,95,105)(92,108,96,112)(121,151,125,147)(122,150,126,146)(123,149,127,145)(124,148,128,152) );
G=PermutationGroup([[(1,78),(2,79),(3,80),(4,73),(5,74),(6,75),(7,76),(8,77),(9,151),(10,152),(11,145),(12,146),(13,147),(14,148),(15,149),(16,150),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(41,120),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,100),(50,101),(51,102),(52,103),(53,104),(54,97),(55,98),(56,99),(57,108),(58,109),(59,110),(60,111),(61,112),(62,105),(63,106),(64,107),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(137,155),(138,156),(139,157),(140,158),(141,159),(142,160),(143,153),(144,154)], [(1,49,39,146,63),(2,64,147,40,50),(3,51,33,148,57),(4,58,149,34,52),(5,53,35,150,59),(6,60,151,36,54),(7,55,37,152,61),(8,62,145,38,56),(9,21,97,75,111),(10,112,76,98,22),(11,23,99,77,105),(12,106,78,100,24),(13,17,101,79,107),(14,108,80,102,18),(15,19,103,73,109),(16,110,74,104,20),(25,143,86,94,48),(26,41,95,87,144),(27,137,88,96,42),(28,43,89,81,138),(29,139,82,90,44),(30,45,91,83,140),(31,141,84,92,46),(32,47,93,85,142),(65,153,126,134,119),(66,120,135,127,154),(67,155,128,136,113),(68,114,129,121,156),(69,157,122,130,115),(70,116,131,123,158),(71,159,124,132,117),(72,118,133,125,160)], [(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)], [(1,115,5,119),(2,114,6,118),(3,113,7,117),(4,120,8,116),(9,85,13,81),(10,84,14,88),(11,83,15,87),(12,82,16,86),(17,138,21,142),(18,137,22,141),(19,144,23,140),(20,143,24,139),(25,100,29,104),(26,99,30,103),(27,98,31,102),(28,97,32,101),(33,155,37,159),(34,154,38,158),(35,153,39,157),(36,160,40,156),(41,77,45,73),(42,76,46,80),(43,75,47,79),(44,74,48,78),(49,69,53,65),(50,68,54,72),(51,67,55,71),(52,66,56,70),(57,136,61,132),(58,135,62,131),(59,134,63,130),(60,133,64,129),(89,111,93,107),(90,110,94,106),(91,109,95,105),(92,108,96,112),(121,151,125,147),(122,150,126,146),(123,149,127,145),(124,148,128,152)]])
C2×C5⋊Q16 is a maximal subgroup of
D20.7D4 C5⋊Q16⋊5C4 Dic5⋊4Q16 Dic5⋊Q16 Dic10.11D4 D10⋊4Q16 Q8.D20 D10⋊Q16 C5⋊2C8.D4 Q8.1D20 C42.59D10 C20⋊7Q16 D20.37D4 Dic10.37D4 (C2×C10)⋊Q16 C5⋊(C8.D4) C42.61D10 C42.214D10 C42.65D10 C42.80D10 C20⋊Q16 C20⋊3Q16 (C5×Q8).D4 C40.31D4 C40.43D4 Dic10.16D4 C40.26D4 Dic5⋊3Q16 D10⋊5Q16 C40.37D4 M4(2).16D10 (C2×C10)⋊8Q16 (C5×D4).32D4 C2×D5×Q16 D20.44D4 D20.35C23
C2×C5⋊Q16 is a maximal quotient of
C4⋊C4.230D10 C20.23Q16 C20⋊7Q16 (C2×C10).Q16 Dic10.37D4 (C2×C10)⋊Q16 C20.17D8 C20.Q16 C20⋊Q16 Dic10⋊5Q8 C20⋊3Q16 C20.11Q16 (C2×C10)⋊8Q16
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | Q16 | D10 | D10 | C5⋊D4 | C5⋊D4 | C5⋊Q16 |
| kernel | C2×C5⋊Q16 | C2×C5⋊2C8 | C5⋊Q16 | C2×Dic10 | Q8×C10 | C20 | C2×C10 | C2×Q8 | C10 | C2×C4 | Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of C2×C5⋊Q16 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 29 | 12 | 0 | 0 |
| 0 | 29 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,1,0,0,0,40,0],[1,0,0,0,0,0,29,29,0,0,0,12,29,0,0,0,0,0,1,0,0,0,0,7,40],[1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1] >;
C2×C5⋊Q16 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes Q_{16} % in TeX
G:=Group("C2xC5:Q16"); // GroupNames label
G:=SmallGroup(160,164);
// by ID
G=gap.SmallGroup(160,164);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,86,579,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations