direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22⋊C16, C22⋊C80, C40.106D4, C23.2C40, C10.14M5(2), C20.60M4(2), (C2×C80)⋊5C2, (C2×C16)⋊1C10, (C2×C10)⋊3C16, (C2×C8).6C20, C2.1(C2×C80), (C2×C4).3C40, C8.26(C5×D4), (C2×C20).16C8, (C2×C40).40C4, C10.21(C2×C16), (C22×C4).8C20, C22.9(C2×C40), (C22×C8).4C10, (C22×C40).7C2, (C22×C10).7C8, C2.2(C5×M5(2)), (C22×C20).43C4, C4.10(C5×M4(2)), C10.40(C22⋊C8), (C2×C40).451C22, C20.158(C22⋊C4), C2.2(C5×C22⋊C8), (C2×C4).83(C2×C20), (C2×C10).68(C2×C8), C4.28(C5×C22⋊C4), (C2×C20).518(C2×C4), (C2×C8).105(C2×C10), SmallGroup(320,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22⋊C16
G = < a,b,c,d | a5=b2=c2=d16=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 90 in 66 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, C22×C8, C40, C40, C2×C20, C2×C20, C22×C10, C22⋊C16, C80, C2×C40, C2×C40, C22×C20, C2×C80, C22×C40, C5×C22⋊C16
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C10, C16, C22⋊C4, C2×C8, M4(2), C20, C2×C10, C22⋊C8, C2×C16, M5(2), C40, C2×C20, C5×D4, C22⋊C16, C80, C5×C22⋊C4, C2×C40, C5×M4(2), C5×C22⋊C8, C2×C80, C5×M5(2), C5×C22⋊C16
►On 160 points
Generators in S160
(1 74 37 89 140)(2 75 38 90 141)(3 76 39 91 142)(4 77 40 92 143)(5 78 41 93 144)(6 79 42 94 129)(7 80 43 95 130)(8 65 44 96 131)(9 66 45 81 132)(10 67 46 82 133)(11 68 47 83 134)(12 69 48 84 135)(13 70 33 85 136)(14 71 34 86 137)(15 72 35 87 138)(16 73 36 88 139)(17 159 111 125 49)(18 160 112 126 50)(19 145 97 127 51)(20 146 98 128 52)(21 147 99 113 53)(22 148 100 114 54)(23 149 101 115 55)(24 150 102 116 56)(25 151 103 117 57)(26 152 104 118 58)(27 153 105 119 59)(28 154 106 120 60)(29 155 107 121 61)(30 156 108 122 62)(31 157 109 123 63)(32 158 110 124 64)
(2 30)(4 32)(6 18)(8 20)(10 22)(12 24)(14 26)(16 28)(34 104)(36 106)(38 108)(40 110)(42 112)(44 98)(46 100)(48 102)(50 129)(52 131)(54 133)(56 135)(58 137)(60 139)(62 141)(64 143)(65 146)(67 148)(69 150)(71 152)(73 154)(75 156)(77 158)(79 160)(82 114)(84 116)(86 118)(88 120)(90 122)(92 124)(94 126)(96 128)
(1 29)(2 30)(3 31)(4 32)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 144)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 145)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)
(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)
G:=sub<Sym(160)| (1,74,37,89,140)(2,75,38,90,141)(3,76,39,91,142)(4,77,40,92,143)(5,78,41,93,144)(6,79,42,94,129)(7,80,43,95,130)(8,65,44,96,131)(9,66,45,81,132)(10,67,46,82,133)(11,68,47,83,134)(12,69,48,84,135)(13,70,33,85,136)(14,71,34,86,137)(15,72,35,87,138)(16,73,36,88,139)(17,159,111,125,49)(18,160,112,126,50)(19,145,97,127,51)(20,146,98,128,52)(21,147,99,113,53)(22,148,100,114,54)(23,149,101,115,55)(24,150,102,116,56)(25,151,103,117,57)(26,152,104,118,58)(27,153,105,119,59)(28,154,106,120,60)(29,155,107,121,61)(30,156,108,122,62)(31,157,109,123,63)(32,158,110,124,64), (2,30)(4,32)(6,18)(8,20)(10,22)(12,24)(14,26)(16,28)(34,104)(36,106)(38,108)(40,110)(42,112)(44,98)(46,100)(48,102)(50,129)(52,131)(54,133)(56,135)(58,137)(60,139)(62,141)(64,143)(65,146)(67,148)(69,150)(71,152)(73,154)(75,156)(77,158)(79,160)(82,114)(84,116)(86,118)(88,120)(90,122)(92,124)(94,126)(96,128), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,144)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,145)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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)>;
G:=Group( (1,74,37,89,140)(2,75,38,90,141)(3,76,39,91,142)(4,77,40,92,143)(5,78,41,93,144)(6,79,42,94,129)(7,80,43,95,130)(8,65,44,96,131)(9,66,45,81,132)(10,67,46,82,133)(11,68,47,83,134)(12,69,48,84,135)(13,70,33,85,136)(14,71,34,86,137)(15,72,35,87,138)(16,73,36,88,139)(17,159,111,125,49)(18,160,112,126,50)(19,145,97,127,51)(20,146,98,128,52)(21,147,99,113,53)(22,148,100,114,54)(23,149,101,115,55)(24,150,102,116,56)(25,151,103,117,57)(26,152,104,118,58)(27,153,105,119,59)(28,154,106,120,60)(29,155,107,121,61)(30,156,108,122,62)(31,157,109,123,63)(32,158,110,124,64), (2,30)(4,32)(6,18)(8,20)(10,22)(12,24)(14,26)(16,28)(34,104)(36,106)(38,108)(40,110)(42,112)(44,98)(46,100)(48,102)(50,129)(52,131)(54,133)(56,135)(58,137)(60,139)(62,141)(64,143)(65,146)(67,148)(69,150)(71,152)(73,154)(75,156)(77,158)(79,160)(82,114)(84,116)(86,118)(88,120)(90,122)(92,124)(94,126)(96,128), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,144)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,145)(81,113)(82,114)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128), (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) );
G=PermutationGroup([[(1,74,37,89,140),(2,75,38,90,141),(3,76,39,91,142),(4,77,40,92,143),(5,78,41,93,144),(6,79,42,94,129),(7,80,43,95,130),(8,65,44,96,131),(9,66,45,81,132),(10,67,46,82,133),(11,68,47,83,134),(12,69,48,84,135),(13,70,33,85,136),(14,71,34,86,137),(15,72,35,87,138),(16,73,36,88,139),(17,159,111,125,49),(18,160,112,126,50),(19,145,97,127,51),(20,146,98,128,52),(21,147,99,113,53),(22,148,100,114,54),(23,149,101,115,55),(24,150,102,116,56),(25,151,103,117,57),(26,152,104,118,58),(27,153,105,119,59),(28,154,106,120,60),(29,155,107,121,61),(30,156,108,122,62),(31,157,109,123,63),(32,158,110,124,64)], [(2,30),(4,32),(6,18),(8,20),(10,22),(12,24),(14,26),(16,28),(34,104),(36,106),(38,108),(40,110),(42,112),(44,98),(46,100),(48,102),(50,129),(52,131),(54,133),(56,135),(58,137),(60,139),(62,141),(64,143),(65,146),(67,148),(69,150),(71,152),(73,154),(75,156),(77,158),(79,160),(82,114),(84,116),(86,118),(88,120),(90,122),(92,124),(94,126),(96,128)], [(1,29),(2,30),(3,31),(4,32),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,111),(42,112),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,144),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,136),(58,137),(59,138),(60,139),(61,140),(62,141),(63,142),(64,143),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,145),(81,113),(82,114),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128)], [(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)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 16A | ··· | 16P | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF | 40AG | ··· | 40AV | 80A | ··· | 80BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C16 | C20 | C20 | C40 | C40 | C80 | D4 | M4(2) | M5(2) | C5×D4 | C5×M4(2) | C5×M5(2) |
| kernel | C5×C22⋊C16 | C2×C80 | C22×C40 | C2×C40 | C22×C20 | C22⋊C16 | C2×C20 | C22×C10 | C2×C16 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C40 | C20 | C10 | C8 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 16 | 8 | 8 | 16 | 16 | 64 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of C5×C22⋊C16 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 240 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 203 | 240 |
| 1 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 76 | 0 | 0 |
| 0 | 222 | 240 |
| 0 | 121 | 19 |
G:=sub<GL(3,GF(241))| [1,0,0,0,98,0,0,0,98],[240,0,0,0,1,203,0,0,240],[1,0,0,0,240,0,0,0,240],[76,0,0,0,222,121,0,240,19] >;
C5×C22⋊C16 in GAP, Magma, Sage, TeX
C_5\times C_2^2\rtimes C_{16} % in TeX
G:=Group("C5xC2^2:C16"); // GroupNames label
G:=SmallGroup(320,153);
// by ID
G=gap.SmallGroup(320,153);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations