direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C8.C22, C40.50C23, C20.83C24, Q16⋊3(C2×C10), C4.67(D4×C10), (C2×Q16)⋊11C10, (C10×Q16)⋊25C2, SD16⋊2(C2×C10), (C2×SD16)⋊5C10, (C2×C20).526D4, C20.330(C2×D4), C8.1(C22×C10), C4.6(C23×C10), (C22×Q8)⋊9C10, C23.51(C5×D4), (C10×SD16)⋊16C2, (C2×M4(2))⋊4C10, M4(2)⋊4(C2×C10), (Q8×C10)⋊55C22, (C5×Q16)⋊17C22, D4.3(C22×C10), (C5×D4).36C23, C22.24(D4×C10), (C5×Q8).37C23, Q8.3(C22×C10), (C10×M4(2))⋊14C2, (C2×C40).280C22, (C2×C20).976C23, (C5×SD16)⋊18C22, (C22×C10).173D4, C10.204(C22×D4), (D4×C10).329C22, (C5×M4(2))⋊30C22, (C22×C20).466C22, (Q8×C2×C10)⋊21C2, C2.28(D4×C2×C10), (C2×C8).32(C2×C10), (C2×Q8)⋊15(C2×C10), (C2×C4).137(C5×D4), (C10×C4○D4).26C2, (C2×C4○D4).12C10, C4○D4.13(C2×C10), (C2×D4).75(C2×C10), (C2×C10).420(C2×D4), (C22×C4).77(C2×C10), (C2×C4).46(C22×C10), (C5×C4○D4).58C22, SmallGroup(320,1576)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 370 in 258 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C10, C10 [×2], C10 [×4], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C20 [×2], C20 [×2], C20 [×6], C2×C10, C2×C10 [×2], C2×C10 [×6], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×11], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×6], C5×Q8 [×7], C22×C10, C22×C10, C2×C8.C22, C2×C40 [×2], C5×M4(2) [×4], C5×SD16 [×8], C5×Q16 [×8], C22×C20, C22×C20 [×2], D4×C10, D4×C10, Q8×C10, Q8×C10 [×6], Q8×C10 [×3], C5×C4○D4 [×4], C5×C4○D4 [×2], C10×M4(2), C10×SD16 [×2], C10×Q16 [×2], C5×C8.C22 [×8], Q8×C2×C10, C10×C4○D4, C10×C8.C22
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C8.C22 [×2], C22×D4, C5×D4 [×4], C22×C10 [×15], C2×C8.C22, D4×C10 [×6], C23×C10, C5×C8.C22 [×2], D4×C2×C10, C10×C8.C22
Generators and relations
G = < a,b,c,d | a10=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
►On 160 points
Generators in S160
(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)
(1 135 62 125 54 119 50 147)(2 136 63 126 55 120 41 148)(3 137 64 127 56 111 42 149)(4 138 65 128 57 112 43 150)(5 139 66 129 58 113 44 141)(6 140 67 130 59 114 45 142)(7 131 68 121 60 115 46 143)(8 132 69 122 51 116 47 144)(9 133 70 123 52 117 48 145)(10 134 61 124 53 118 49 146)(11 75 39 81 29 91 160 103)(12 76 40 82 30 92 151 104)(13 77 31 83 21 93 152 105)(14 78 32 84 22 94 153 106)(15 79 33 85 23 95 154 107)(16 80 34 86 24 96 155 108)(17 71 35 87 25 97 156 109)(18 72 36 88 26 98 157 110)(19 73 37 89 27 99 158 101)(20 74 38 90 28 100 159 102)
(11 39)(12 40)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 61)(50 62)(71 97)(72 98)(73 99)(74 100)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 141)(28 142)(29 143)(30 144)(31 117)(32 118)(33 119)(34 120)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 108)(42 109)(43 110)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 91)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 81)(69 82)(70 83)(131 160)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 157)(139 158)(140 159)
G:=sub<Sym(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), (1,135,62,125,54,119,50,147)(2,136,63,126,55,120,41,148)(3,137,64,127,56,111,42,149)(4,138,65,128,57,112,43,150)(5,139,66,129,58,113,44,141)(6,140,67,130,59,114,45,142)(7,131,68,121,60,115,46,143)(8,132,69,122,51,116,47,144)(9,133,70,123,52,117,48,145)(10,134,61,124,53,118,49,146)(11,75,39,81,29,91,160,103)(12,76,40,82,30,92,151,104)(13,77,31,83,21,93,152,105)(14,78,32,84,22,94,153,106)(15,79,33,85,23,95,154,107)(16,80,34,86,24,96,155,108)(17,71,35,87,25,97,156,109)(18,72,36,88,26,98,157,110)(19,73,37,89,27,99,158,101)(20,74,38,90,28,100,159,102), (11,39)(12,40)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,61)(50,62)(71,97)(72,98)(73,99)(74,100)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,141)(28,142)(29,143)(30,144)(31,117)(32,118)(33,119)(34,120)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,108)(42,109)(43,110)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,91)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,81)(69,82)(70,83)(131,160)(132,151)(133,152)(134,153)(135,154)(136,155)(137,156)(138,157)(139,158)(140,159)>;
G:=Group( (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), (1,135,62,125,54,119,50,147)(2,136,63,126,55,120,41,148)(3,137,64,127,56,111,42,149)(4,138,65,128,57,112,43,150)(5,139,66,129,58,113,44,141)(6,140,67,130,59,114,45,142)(7,131,68,121,60,115,46,143)(8,132,69,122,51,116,47,144)(9,133,70,123,52,117,48,145)(10,134,61,124,53,118,49,146)(11,75,39,81,29,91,160,103)(12,76,40,82,30,92,151,104)(13,77,31,83,21,93,152,105)(14,78,32,84,22,94,153,106)(15,79,33,85,23,95,154,107)(16,80,34,86,24,96,155,108)(17,71,35,87,25,97,156,109)(18,72,36,88,26,98,157,110)(19,73,37,89,27,99,158,101)(20,74,38,90,28,100,159,102), (11,39)(12,40)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,61)(50,62)(71,97)(72,98)(73,99)(74,100)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,121)(12,122)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,141)(28,142)(29,143)(30,144)(31,117)(32,118)(33,119)(34,120)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,108)(42,109)(43,110)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,91)(61,84)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,81)(69,82)(70,83)(131,160)(132,151)(133,152)(134,153)(135,154)(136,155)(137,156)(138,157)(139,158)(140,159) );
G=PermutationGroup([(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)], [(1,135,62,125,54,119,50,147),(2,136,63,126,55,120,41,148),(3,137,64,127,56,111,42,149),(4,138,65,128,57,112,43,150),(5,139,66,129,58,113,44,141),(6,140,67,130,59,114,45,142),(7,131,68,121,60,115,46,143),(8,132,69,122,51,116,47,144),(9,133,70,123,52,117,48,145),(10,134,61,124,53,118,49,146),(11,75,39,81,29,91,160,103),(12,76,40,82,30,92,151,104),(13,77,31,83,21,93,152,105),(14,78,32,84,22,94,153,106),(15,79,33,85,23,95,154,107),(16,80,34,86,24,96,155,108),(17,71,35,87,25,97,156,109),(18,72,36,88,26,98,157,110),(19,73,37,89,27,99,158,101),(20,74,38,90,28,100,159,102)], [(11,39),(12,40),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,151),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,61),(50,62),(71,97),(72,98),(73,99),(74,100),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(111,149),(112,150),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140)], [(1,79),(2,80),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,121),(12,122),(13,123),(14,124),(15,125),(16,126),(17,127),(18,128),(19,129),(20,130),(21,145),(22,146),(23,147),(24,148),(25,149),(26,150),(27,141),(28,142),(29,143),(30,144),(31,117),(32,118),(33,119),(34,120),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,108),(42,109),(43,110),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,91),(61,84),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,81),(69,82),(70,83),(131,160),(132,151),(133,152),(134,153),(135,154),(136,155),(137,156),(138,157),(139,158),(140,159)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 31 | 0 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 10 | 37 | 0 | 0 | 0 | 0 |
| 15 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 27 | 20 | 20 |
| 0 | 0 | 14 | 34 | 21 | 20 |
| 0 | 0 | 7 | 14 | 7 | 14 |
| 0 | 0 | 27 | 7 | 27 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [31,0,0,0,0,0,0,31,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[10,15,0,0,0,0,37,31,0,0,0,0,0,0,34,14,7,27,0,0,27,34,14,7,0,0,20,21,7,27,0,0,20,20,14,7],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,1,0,40,0,0,0,0,40,0,1,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,0,40,0,0,0,0,2,0,40] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AN | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | C8.C22 | C5×C8.C22 |
| kernel | C10×C8.C22 | C10×M4(2) | C10×SD16 | C10×Q16 | C5×C8.C22 | Q8×C2×C10 | C10×C4○D4 | C2×C8.C22 | C2×M4(2) | C2×SD16 | C2×Q16 | C8.C22 | C22×Q8 | C2×C4○D4 | C2×C20 | C22×C10 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 4 | 4 | 8 | 8 | 32 | 4 | 4 | 3 | 1 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times C_8.C_2^2 % in TeX
G:=Group("C10xC8.C2^2"); // GroupNames label
G:=SmallGroup(320,1576);
// by ID
G=gap.SmallGroup(320,1576);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀