direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C2×C10, C40⋊15C23, C24.5C20, C20.92C24, C8⋊4(C22×C10), (C22×C40)⋊26C2, (C2×C40)⋊53C22, (C22×C8)⋊12C10, (C22×C20).67C4, C10.83(C23×C4), C4.16(C23×C10), C23.40(C2×C20), (C22×C4).18C20, C2.10(C23×C20), C4.31(C22×C20), (C23×C20).26C2, (C23×C4).11C10, (C23×C10).13C4, (C2×C20).968C23, C20.248(C22×C4), C22.27(C22×C20), (C22×C20).599C22, (C2×C8)⋊15(C2×C10), (C2×C4).79(C2×C20), (C2×C20).514(C2×C4), (C2×C10).267(C22×C4), (C2×C4).138(C22×C10), (C22×C10).189(C2×C4), (C22×C4).126(C2×C10), SmallGroup(320,1568)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 338 in 298 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×7], C22 [×11], C22 [×12], C5, C8 [×8], C2×C4 [×28], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C2×C8 [×12], M4(2) [×16], C22×C4 [×2], C22×C4 [×12], C24, C20, C20 [×7], C2×C10 [×11], C2×C10 [×12], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C40 [×8], C2×C20 [×28], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×M4(2), C2×C40 [×12], C5×M4(2) [×16], C22×C20 [×2], C22×C20 [×12], C23×C10, C22×C40 [×2], C10×M4(2) [×12], C23×C20, M4(2)×C2×C10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], M4(2) [×4], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C2×M4(2) [×6], C23×C4, C2×C20 [×28], C22×C10 [×15], C22×M4(2), C5×M4(2) [×4], C22×C20 [×14], C23×C10, C10×M4(2) [×6], C23×C20, M4(2)×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=c5 >
►On 160 points
Generators in S160
(1 108)(2 109)(3 110)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 137)(12 138)(13 139)(14 140)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 118)(22 119)(23 120)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 86)(52 87)(53 88)(54 89)(55 90)(56 81)(57 82)(58 83)(59 84)(60 85)(61 100)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(121 159)(122 160)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)
(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)
(1 154 51 16 63 22 41 38)(2 155 52 17 64 23 42 39)(3 156 53 18 65 24 43 40)(4 157 54 19 66 25 44 31)(5 158 55 20 67 26 45 32)(6 159 56 11 68 27 46 33)(7 160 57 12 69 28 47 34)(8 151 58 13 70 29 48 35)(9 152 59 14 61 30 49 36)(10 153 60 15 62 21 50 37)(71 150 110 128 88 134 94 111)(72 141 101 129 89 135 95 112)(73 142 102 130 90 136 96 113)(74 143 103 121 81 137 97 114)(75 144 104 122 82 138 98 115)(76 145 105 123 83 139 99 116)(77 146 106 124 84 140 100 117)(78 147 107 125 85 131 91 118)(79 148 108 126 86 132 92 119)(80 149 109 127 87 133 93 120)
(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 31)(20 32)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 151)(30 152)(111 128)(112 129)(113 130)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
G:=sub<Sym(160)| (1,108)(2,109)(3,110)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,137)(12,138)(13,139)(14,140)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,118)(22,119)(23,120)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,86)(52,87)(53,88)(54,89)(55,90)(56,81)(57,82)(58,83)(59,84)(60,85)(61,100)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158), (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), (1,154,51,16,63,22,41,38)(2,155,52,17,64,23,42,39)(3,156,53,18,65,24,43,40)(4,157,54,19,66,25,44,31)(5,158,55,20,67,26,45,32)(6,159,56,11,68,27,46,33)(7,160,57,12,69,28,47,34)(8,151,58,13,70,29,48,35)(9,152,59,14,61,30,49,36)(10,153,60,15,62,21,50,37)(71,150,110,128,88,134,94,111)(72,141,101,129,89,135,95,112)(73,142,102,130,90,136,96,113)(74,143,103,121,81,137,97,114)(75,144,104,122,82,138,98,115)(76,145,105,123,83,139,99,116)(77,146,106,124,84,140,100,117)(78,147,107,125,85,131,91,118)(79,148,108,126,86,132,92,119)(80,149,109,127,87,133,93,120), (11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,31)(20,32)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,151)(30,152)(111,128)(112,129)(113,130)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)>;
G:=Group( (1,108)(2,109)(3,110)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,137)(12,138)(13,139)(14,140)(15,131)(16,132)(17,133)(18,134)(19,135)(20,136)(21,118)(22,119)(23,120)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,86)(52,87)(53,88)(54,89)(55,90)(56,81)(57,82)(58,83)(59,84)(60,85)(61,100)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,99)(121,159)(122,160)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158), (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), (1,154,51,16,63,22,41,38)(2,155,52,17,64,23,42,39)(3,156,53,18,65,24,43,40)(4,157,54,19,66,25,44,31)(5,158,55,20,67,26,45,32)(6,159,56,11,68,27,46,33)(7,160,57,12,69,28,47,34)(8,151,58,13,70,29,48,35)(9,152,59,14,61,30,49,36)(10,153,60,15,62,21,50,37)(71,150,110,128,88,134,94,111)(72,141,101,129,89,135,95,112)(73,142,102,130,90,136,96,113)(74,143,103,121,81,137,97,114)(75,144,104,122,82,138,98,115)(76,145,105,123,83,139,99,116)(77,146,106,124,84,140,100,117)(78,147,107,125,85,131,91,118)(79,148,108,126,86,132,92,119)(80,149,109,127,87,133,93,120), (11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,31)(20,32)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,151)(30,152)(111,128)(112,129)(113,130)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(120,127)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146) );
G=PermutationGroup([(1,108),(2,109),(3,110),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,137),(12,138),(13,139),(14,140),(15,131),(16,132),(17,133),(18,134),(19,135),(20,136),(21,118),(22,119),(23,120),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,141),(32,142),(33,143),(34,144),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,79),(42,80),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,86),(52,87),(53,88),(54,89),(55,90),(56,81),(57,82),(58,83),(59,84),(60,85),(61,100),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,99),(121,159),(122,160),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158)], [(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)], [(1,154,51,16,63,22,41,38),(2,155,52,17,64,23,42,39),(3,156,53,18,65,24,43,40),(4,157,54,19,66,25,44,31),(5,158,55,20,67,26,45,32),(6,159,56,11,68,27,46,33),(7,160,57,12,69,28,47,34),(8,151,58,13,70,29,48,35),(9,152,59,14,61,30,49,36),(10,153,60,15,62,21,50,37),(71,150,110,128,88,134,94,111),(72,141,101,129,89,135,95,112),(73,142,102,130,90,136,96,113),(74,143,103,121,81,137,97,114),(75,144,104,122,82,138,98,115),(76,145,105,123,83,139,99,116),(77,146,106,124,84,140,100,117),(78,147,107,125,85,131,91,118),(79,148,108,126,86,132,92,119),(80,149,109,127,87,133,93,120)], [(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,31),(20,32),(21,153),(22,154),(23,155),(24,156),(25,157),(26,158),(27,159),(28,160),(29,151),(30,152),(111,128),(112,129),(113,130),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(120,127),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 39 | 2 |
| 0 | 0 | 14 | 2 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,31,0,0,0,0,31],[40,0,0,0,0,1,0,0,0,0,39,14,0,0,2,2],[40,0,0,0,0,40,0,0,0,0,1,2,0,0,0,40] >;
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20AV | 40A | ··· | 40BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 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 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | M4(2) | C5×M4(2) |
| kernel | M4(2)×C2×C10 | C22×C40 | C10×M4(2) | C23×C20 | C22×C20 | C23×C10 | C22×M4(2) | C22×C8 | C2×M4(2) | C23×C4 | C22×C4 | C24 | C2×C10 | C22 |
| # reps | 1 | 2 | 12 | 1 | 14 | 2 | 4 | 8 | 48 | 4 | 56 | 8 | 8 | 32 |
In GAP, Magma, Sage, TeX
M_{4(2)}\times C_2\times C_{10} % in TeX
G:=Group("M4(2)xC2xC10"); // GroupNames label
G:=SmallGroup(320,1568);
// by ID
G=gap.SmallGroup(320,1568);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,2269,124]);
// 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^5>;
// generators/relations
×
⋊
ℤ
𝔽
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