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), (C10×Q16)⋊25C2, (C2×Q16)⋊11C10, (C2×SD16)⋊5C10, SD16⋊2(C2×C10), (C2×C20).526D4, C20.330(C2×D4), C4.6(C23×C10), C8.1(C22×C10), (C22×Q8)⋊9C10, C23.51(C5×D4), (C10×SD16)⋊16C2, (C2×M4(2))⋊4C10, M4(2)⋊4(C2×C10), (C5×Q16)⋊17C22, (Q8×C10)⋊55C22, (C5×D4).36C23, D4.3(C22×C10), C22.24(D4×C10), Q8.3(C22×C10), (C5×Q8).37C23, (C10×M4(2))⋊14C2, (C2×C20).976C23, (C2×C40).280C22, (C5×SD16)⋊18C22, C10.204(C22×D4), (C22×C10).173D4, (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), (C2×C4○D4).12C10, (C10×C4○D4).26C2, C4○D4.13(C2×C10), (C2×D4).75(C2×C10), (C2×C10).420(C2×D4), (C2×C4).46(C22×C10), (C22×C4).77(C2×C10), (C5×C4○D4).58C22, SmallGroup(320,1576)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C8.C22
G = < a,b,c,d | a10=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 370 in 258 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C2×C8.C22, C2×C40, C5×M4(2), C5×SD16, C5×Q16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C10×M4(2), C10×SD16, C10×Q16, C5×C8.C22, Q8×C2×C10, C10×C4○D4, C10×C8.C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C8.C22, C22×D4, C5×D4, C22×C10, C2×C8.C22, D4×C10, C23×C10, C5×C8.C22, D4×C2×C10, C10×C8.C22
►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)
(1 142 67 125 60 119 50 138)(2 143 68 126 51 120 41 139)(3 144 69 127 52 111 42 140)(4 145 70 128 53 112 43 131)(5 146 61 129 54 113 44 132)(6 147 62 130 55 114 45 133)(7 148 63 121 56 115 46 134)(8 149 64 122 57 116 47 135)(9 150 65 123 58 117 48 136)(10 141 66 124 59 118 49 137)(11 90 28 107 159 93 31 74)(12 81 29 108 160 94 32 75)(13 82 30 109 151 95 33 76)(14 83 21 110 152 96 34 77)(15 84 22 101 153 97 35 78)(16 85 23 102 154 98 36 79)(17 86 24 103 155 99 37 80)(18 87 25 104 156 100 38 71)(19 88 26 105 157 91 39 72)(20 89 27 106 158 92 40 73)
(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(41 68)(42 69)(43 70)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 101)(79 102)(80 103)(111 140)(112 131)(113 132)(114 133)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 148)(122 149)(123 150)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 111)(19 112)(20 113)(21 136)(22 137)(23 138)(24 139)(25 140)(26 131)(27 132)(28 133)(29 134)(30 135)(31 130)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 99)(42 100)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 101)(60 102)(61 89)(62 90)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(141 153)(142 154)(143 155)(144 156)(145 157)(146 158)(147 159)(148 160)(149 151)(150 152)
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), (1,142,67,125,60,119,50,138)(2,143,68,126,51,120,41,139)(3,144,69,127,52,111,42,140)(4,145,70,128,53,112,43,131)(5,146,61,129,54,113,44,132)(6,147,62,130,55,114,45,133)(7,148,63,121,56,115,46,134)(8,149,64,122,57,116,47,135)(9,150,65,123,58,117,48,136)(10,141,66,124,59,118,49,137)(11,90,28,107,159,93,31,74)(12,81,29,108,160,94,32,75)(13,82,30,109,151,95,33,76)(14,83,21,110,152,96,34,77)(15,84,22,101,153,97,35,78)(16,85,23,102,154,98,36,79)(17,86,24,103,155,99,37,80)(18,87,25,104,156,100,38,71)(19,88,26,105,157,91,39,72)(20,89,27,106,158,92,40,73), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(41,68)(42,69)(43,70)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,101)(79,102)(80,103)(111,140)(112,131)(113,132)(114,133)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,148)(122,149)(123,150)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,111)(19,112)(20,113)(21,136)(22,137)(23,138)(24,139)(25,140)(26,131)(27,132)(28,133)(29,134)(30,135)(31,130)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,99)(42,100)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(141,153)(142,154)(143,155)(144,156)(145,157)(146,158)(147,159)(148,160)(149,151)(150,152)>;
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), (1,142,67,125,60,119,50,138)(2,143,68,126,51,120,41,139)(3,144,69,127,52,111,42,140)(4,145,70,128,53,112,43,131)(5,146,61,129,54,113,44,132)(6,147,62,130,55,114,45,133)(7,148,63,121,56,115,46,134)(8,149,64,122,57,116,47,135)(9,150,65,123,58,117,48,136)(10,141,66,124,59,118,49,137)(11,90,28,107,159,93,31,74)(12,81,29,108,160,94,32,75)(13,82,30,109,151,95,33,76)(14,83,21,110,152,96,34,77)(15,84,22,101,153,97,35,78)(16,85,23,102,154,98,36,79)(17,86,24,103,155,99,37,80)(18,87,25,104,156,100,38,71)(19,88,26,105,157,91,39,72)(20,89,27,106,158,92,40,73), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(41,68)(42,69)(43,70)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,101)(79,102)(80,103)(111,140)(112,131)(113,132)(114,133)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,148)(122,149)(123,150)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,111)(19,112)(20,113)(21,136)(22,137)(23,138)(24,139)(25,140)(26,131)(27,132)(28,133)(29,134)(30,135)(31,130)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,99)(42,100)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(141,153)(142,154)(143,155)(144,156)(145,157)(146,158)(147,159)(148,160)(149,151)(150,152) );
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)], [(1,142,67,125,60,119,50,138),(2,143,68,126,51,120,41,139),(3,144,69,127,52,111,42,140),(4,145,70,128,53,112,43,131),(5,146,61,129,54,113,44,132),(6,147,62,130,55,114,45,133),(7,148,63,121,56,115,46,134),(8,149,64,122,57,116,47,135),(9,150,65,123,58,117,48,136),(10,141,66,124,59,118,49,137),(11,90,28,107,159,93,31,74),(12,81,29,108,160,94,32,75),(13,82,30,109,151,95,33,76),(14,83,21,110,152,96,34,77),(15,84,22,101,153,97,35,78),(16,85,23,102,154,98,36,79),(17,86,24,103,155,99,37,80),(18,87,25,104,156,100,38,71),(19,88,26,105,157,91,39,72),(20,89,27,106,158,92,40,73)], [(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,151),(41,68),(42,69),(43,70),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,101),(79,102),(80,103),(111,140),(112,131),(113,132),(114,133),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,148),(122,149),(123,150),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147)], [(1,79),(2,80),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,111),(19,112),(20,113),(21,136),(22,137),(23,138),(24,139),(25,140),(26,131),(27,132),(28,133),(29,134),(30,135),(31,130),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,99),(42,100),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,98),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,101),(60,102),(61,89),(62,90),(63,81),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(141,153),(142,154),(143,155),(144,156),(145,157),(146,158),(147,159),(148,160),(149,151),(150,152)]])
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 |
Matrix representation of C10×C8.C22 ►in 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] >;
C10×C8.C22 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