direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C2×C10, C40⋊13C23, C20.78C24, C8⋊2(C22×C10), (C22×C8)⋊7C10, C4.18(D4×C10), (C2×C40)⋊50C22, (C22×C40)⋊21C2, (C2×C20).432D4, C20.325(C2×D4), (C5×D4)⋊12C23, D4⋊1(C22×C10), C4.1(C23×C10), C23.60(C5×D4), (D4×C10)⋊65C22, (C22×D4)⋊10C10, C22.65(D4×C10), (C2×C20).971C23, C10.199(C22×D4), (C22×C10).221D4, (C22×C20).601C22, (D4×C2×C10)⋊25C2, C2.23(D4×C2×C10), (C2×C8)⋊12(C2×C10), (C2×C4).88(C5×D4), (C2×D4)⋊14(C2×C10), (C2×C10).686(C2×D4), (C2×C4).141(C22×C10), (C22×C4).128(C2×C10), SmallGroup(320,1571)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 658 in 338 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×3], C22 [×7], C22 [×32], C5, C8 [×4], C2×C4 [×6], D4 [×8], D4 [×12], C23, C23 [×20], C10, C10 [×6], C10 [×8], C2×C8 [×6], D8 [×16], C22×C4, C2×D4 [×12], C2×D4 [×6], C24 [×2], C20, C20 [×3], C2×C10 [×7], C2×C10 [×32], C22×C8, C2×D8 [×12], C22×D4 [×2], C40 [×4], C2×C20 [×6], C5×D4 [×8], C5×D4 [×12], C22×C10, C22×C10 [×20], C22×D8, C2×C40 [×6], C5×D8 [×16], C22×C20, D4×C10 [×12], D4×C10 [×6], C23×C10 [×2], C22×C40, C10×D8 [×12], D4×C2×C10 [×2], D8×C2×C10
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], D8 [×4], C2×D4 [×6], C24, C2×C10 [×35], C2×D8 [×6], C22×D4, C5×D4 [×4], C22×C10 [×15], C22×D8, C5×D8 [×4], D4×C10 [×6], C23×C10, C10×D8 [×6], D4×C2×C10, D8×C2×C10
Generators and relations
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 111)(29 112)(30 113)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 71)(48 72)(49 73)(50 74)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 101)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 81)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 151)(128 152)(129 153)(130 154)
(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 22 65 36 55 160 41 20)(2 23 66 37 56 151 42 11)(3 24 67 38 57 152 43 12)(4 25 68 39 58 153 44 13)(5 26 69 40 59 154 45 14)(6 27 70 31 60 155 46 15)(7 28 61 32 51 156 47 16)(8 29 62 33 52 157 48 17)(9 30 63 34 53 158 49 18)(10 21 64 35 54 159 50 19)(71 136 96 111 82 142 102 122)(72 137 97 112 83 143 103 123)(73 138 98 113 84 144 104 124)(74 139 99 114 85 145 105 125)(75 140 100 115 86 146 106 126)(76 131 91 116 87 147 107 127)(77 132 92 117 88 148 108 128)(78 133 93 118 89 149 109 129)(79 134 94 119 90 150 110 130)(80 135 95 120 81 141 101 121)
(1 20)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(21 50)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 60)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(61 156)(62 157)(63 158)(64 159)(65 160)(66 151)(67 152)(68 153)(69 154)(70 155)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(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)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)
G:=sub<Sym(160)| (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,111)(29,112)(30,113)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,71)(48,72)(49,73)(50,74)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,101)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,81)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,151)(128,152)(129,153)(130,154), (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), (1,22,65,36,55,160,41,20)(2,23,66,37,56,151,42,11)(3,24,67,38,57,152,43,12)(4,25,68,39,58,153,44,13)(5,26,69,40,59,154,45,14)(6,27,70,31,60,155,46,15)(7,28,61,32,51,156,47,16)(8,29,62,33,52,157,48,17)(9,30,63,34,53,158,49,18)(10,21,64,35,54,159,50,19)(71,136,96,111,82,142,102,122)(72,137,97,112,83,143,103,123)(73,138,98,113,84,144,104,124)(74,139,99,114,85,145,105,125)(75,140,100,115,86,146,106,126)(76,131,91,116,87,147,107,127)(77,132,92,117,88,148,108,128)(78,133,93,118,89,149,109,129)(79,134,94,119,90,150,110,130)(80,135,95,120,81,141,101,121), (1,20)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,60)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,156)(62,157)(63,158)(64,159)(65,160)(66,151)(67,152)(68,153)(69,154)(70,155)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(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)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)>;
G:=Group( (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,111)(29,112)(30,113)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,71)(48,72)(49,73)(50,74)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,101)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,81)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(127,151)(128,152)(129,153)(130,154), (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), (1,22,65,36,55,160,41,20)(2,23,66,37,56,151,42,11)(3,24,67,38,57,152,43,12)(4,25,68,39,58,153,44,13)(5,26,69,40,59,154,45,14)(6,27,70,31,60,155,46,15)(7,28,61,32,51,156,47,16)(8,29,62,33,52,157,48,17)(9,30,63,34,53,158,49,18)(10,21,64,35,54,159,50,19)(71,136,96,111,82,142,102,122)(72,137,97,112,83,143,103,123)(73,138,98,113,84,144,104,124)(74,139,99,114,85,145,105,125)(75,140,100,115,86,146,106,126)(76,131,91,116,87,147,107,127)(77,132,92,117,88,148,108,128)(78,133,93,118,89,149,109,129)(79,134,94,119,90,150,110,130)(80,135,95,120,81,141,101,121), (1,20)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(21,50)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,60)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(61,156)(62,157)(63,158)(64,159)(65,160)(66,151)(67,152)(68,153)(69,154)(70,155)(71,111)(72,112)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(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)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150) );
G=PermutationGroup([(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,111),(29,112),(30,113),(31,141),(32,142),(33,143),(34,144),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,71),(48,72),(49,73),(50,74),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,101),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,81),(121,155),(122,156),(123,157),(124,158),(125,159),(126,160),(127,151),(128,152),(129,153),(130,154)], [(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)], [(1,22,65,36,55,160,41,20),(2,23,66,37,56,151,42,11),(3,24,67,38,57,152,43,12),(4,25,68,39,58,153,44,13),(5,26,69,40,59,154,45,14),(6,27,70,31,60,155,46,15),(7,28,61,32,51,156,47,16),(8,29,62,33,52,157,48,17),(9,30,63,34,53,158,49,18),(10,21,64,35,54,159,50,19),(71,136,96,111,82,142,102,122),(72,137,97,112,83,143,103,123),(73,138,98,113,84,144,104,124),(74,139,99,114,85,145,105,125),(75,140,100,115,86,146,106,126),(76,131,91,116,87,147,107,127),(77,132,92,117,88,148,108,128),(78,133,93,118,89,149,109,129),(79,134,94,119,90,150,110,130),(80,135,95,120,81,141,101,121)], [(1,20),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(21,50),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,60),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(61,156),(62,157),(63,158),(64,159),(65,160),(66,151),(67,152),(68,153),(69,154),(70,155),(71,111),(72,112),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(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),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 29 | 29 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,0,0,0,0,0,0,23,0,0,0,0,0,0,25,0,0,0,0,0,0,25,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,29,0,0,0,0,29,29] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 10AC | ··· | 10BH | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | D8 | C5×D4 | C5×D4 | C5×D8 |
| kernel | D8×C2×C10 | C22×C40 | C10×D8 | D4×C2×C10 | C22×D8 | C22×C8 | C2×D8 | C22×D4 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 2 | 4 | 4 | 48 | 8 | 3 | 1 | 8 | 12 | 4 | 32 |
In GAP, Magma, Sage, TeX
D_8\times C_2\times C_{10} % in TeX
G:=Group("D8xC2xC10"); // GroupNames label
G:=SmallGroup(320,1571);
// by ID
G=gap.SmallGroup(320,1571);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=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^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀