direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4⋊M4(2), C20⋊15M4(2), C42.11C20, C4⋊C8⋊11C10, (C4×C20).43C4, C4.75(D4×C10), C4⋊2(C5×M4(2)), C20.91(C4⋊C4), (C2×C20).74Q8, C4.22(Q8×C10), C20.480(C2×D4), (C2×C20).416D4, C20.128(C2×Q8), (C22×C4).15C20, (C22×C20).63C4, C23.31(C2×C20), C42.67(C2×C10), (C2×C42).14C10, C2.7(C10×M4(2)), (C2×C40).324C22, (C4×C20).351C22, (C2×C20).985C23, C10.84(C2×M4(2)), (C10×M4(2)).31C2, (C2×M4(2)).13C10, C22.43(C22×C20), (C22×C20).585C22, (C5×C4⋊C8)⋊30C2, C4.11(C5×C4⋊C4), C2.9(C10×C4⋊C4), (C2×C4×C20).37C2, C10.88(C2×C4⋊C4), (C2×C4).71(C5×D4), C22.7(C5×C4⋊C4), (C2×C4).16(C5×Q8), (C2×C4).73(C2×C20), (C2×C8).49(C2×C10), (C2×C10).52(C4⋊C4), (C2×C20).462(C2×C4), (C2×C10).336(C22×C4), (C22×C10).185(C2×C4), (C2×C4).153(C22×C10), (C22×C4).114(C2×C10), SmallGroup(320,924)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊M4(2)
G = < a,b,c,d | a5=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd=c5 >
Subgroups: 162 in 126 conjugacy classes, 90 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C2×C42, C2×M4(2), C40, C2×C20, C2×C20, C2×C20, C22×C10, C4⋊M4(2), C4×C20, C4×C20, C2×C40, C5×M4(2), C22×C20, C22×C20, C5×C4⋊C8, C2×C4×C20, C10×M4(2), C5×C4⋊M4(2)
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C4⋊C4, C2×M4(2), C2×C20, C5×D4, C5×Q8, C22×C10, C4⋊M4(2), C5×C4⋊C4, C5×M4(2), C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C10×M4(2), C5×C4⋊M4(2)
►On 160 points
Generators in S160
(1 139 63 131 55)(2 140 64 132 56)(3 141 57 133 49)(4 142 58 134 50)(5 143 59 135 51)(6 144 60 136 52)(7 137 61 129 53)(8 138 62 130 54)(9 104 25 17 95)(10 97 26 18 96)(11 98 27 19 89)(12 99 28 20 90)(13 100 29 21 91)(14 101 30 22 92)(15 102 31 23 93)(16 103 32 24 94)(33 111 123 120 41)(34 112 124 113 42)(35 105 125 114 43)(36 106 126 115 44)(37 107 127 116 45)(38 108 128 117 46)(39 109 121 118 47)(40 110 122 119 48)(65 84 157 73 149)(66 85 158 74 150)(67 86 159 75 151)(68 87 160 76 152)(69 88 153 77 145)(70 81 154 78 146)(71 82 155 79 147)(72 83 156 80 148)
(1 65 15 111)(2 112 16 66)(3 67 9 105)(4 106 10 68)(5 69 11 107)(6 108 12 70)(7 71 13 109)(8 110 14 72)(17 43 133 75)(18 76 134 44)(19 45 135 77)(20 78 136 46)(21 47 129 79)(22 80 130 48)(23 41 131 73)(24 74 132 42)(25 114 57 159)(26 160 58 115)(27 116 59 153)(28 154 60 117)(29 118 61 155)(30 156 62 119)(31 120 63 157)(32 158 64 113)(33 55 149 93)(34 94 150 56)(35 49 151 95)(36 96 152 50)(37 51 145 89)(38 90 146 52)(39 53 147 91)(40 92 148 54)(81 144 128 99)(82 100 121 137)(83 138 122 101)(84 102 123 139)(85 140 124 103)(86 104 125 141)(87 142 126 97)(88 98 127 143)
(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 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 129)(18 134)(19 131)(20 136)(21 133)(22 130)(23 135)(24 132)(25 61)(26 58)(27 63)(28 60)(29 57)(30 62)(31 59)(32 64)(33 145)(34 150)(35 147)(36 152)(37 149)(38 146)(39 151)(40 148)(41 77)(42 74)(43 79)(44 76)(45 73)(46 78)(47 75)(48 80)(49 91)(50 96)(51 93)(52 90)(53 95)(54 92)(55 89)(56 94)(65 107)(66 112)(67 109)(68 106)(69 111)(70 108)(71 105)(72 110)(81 128)(82 125)(83 122)(84 127)(85 124)(86 121)(87 126)(88 123)(97 142)(98 139)(99 144)(100 141)(101 138)(102 143)(103 140)(104 137)(113 158)(114 155)(115 160)(116 157)(117 154)(118 159)(119 156)(120 153)
G:=sub<Sym(160)| (1,139,63,131,55)(2,140,64,132,56)(3,141,57,133,49)(4,142,58,134,50)(5,143,59,135,51)(6,144,60,136,52)(7,137,61,129,53)(8,138,62,130,54)(9,104,25,17,95)(10,97,26,18,96)(11,98,27,19,89)(12,99,28,20,90)(13,100,29,21,91)(14,101,30,22,92)(15,102,31,23,93)(16,103,32,24,94)(33,111,123,120,41)(34,112,124,113,42)(35,105,125,114,43)(36,106,126,115,44)(37,107,127,116,45)(38,108,128,117,46)(39,109,121,118,47)(40,110,122,119,48)(65,84,157,73,149)(66,85,158,74,150)(67,86,159,75,151)(68,87,160,76,152)(69,88,153,77,145)(70,81,154,78,146)(71,82,155,79,147)(72,83,156,80,148), (1,65,15,111)(2,112,16,66)(3,67,9,105)(4,106,10,68)(5,69,11,107)(6,108,12,70)(7,71,13,109)(8,110,14,72)(17,43,133,75)(18,76,134,44)(19,45,135,77)(20,78,136,46)(21,47,129,79)(22,80,130,48)(23,41,131,73)(24,74,132,42)(25,114,57,159)(26,160,58,115)(27,116,59,153)(28,154,60,117)(29,118,61,155)(30,156,62,119)(31,120,63,157)(32,158,64,113)(33,55,149,93)(34,94,150,56)(35,49,151,95)(36,96,152,50)(37,51,145,89)(38,90,146,52)(39,53,147,91)(40,92,148,54)(81,144,128,99)(82,100,121,137)(83,138,122,101)(84,102,123,139)(85,140,124,103)(86,104,125,141)(87,142,126,97)(88,98,127,143), (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), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,129)(18,134)(19,131)(20,136)(21,133)(22,130)(23,135)(24,132)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(32,64)(33,145)(34,150)(35,147)(36,152)(37,149)(38,146)(39,151)(40,148)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,91)(50,96)(51,93)(52,90)(53,95)(54,92)(55,89)(56,94)(65,107)(66,112)(67,109)(68,106)(69,111)(70,108)(71,105)(72,110)(81,128)(82,125)(83,122)(84,127)(85,124)(86,121)(87,126)(88,123)(97,142)(98,139)(99,144)(100,141)(101,138)(102,143)(103,140)(104,137)(113,158)(114,155)(115,160)(116,157)(117,154)(118,159)(119,156)(120,153)>;
G:=Group( (1,139,63,131,55)(2,140,64,132,56)(3,141,57,133,49)(4,142,58,134,50)(5,143,59,135,51)(6,144,60,136,52)(7,137,61,129,53)(8,138,62,130,54)(9,104,25,17,95)(10,97,26,18,96)(11,98,27,19,89)(12,99,28,20,90)(13,100,29,21,91)(14,101,30,22,92)(15,102,31,23,93)(16,103,32,24,94)(33,111,123,120,41)(34,112,124,113,42)(35,105,125,114,43)(36,106,126,115,44)(37,107,127,116,45)(38,108,128,117,46)(39,109,121,118,47)(40,110,122,119,48)(65,84,157,73,149)(66,85,158,74,150)(67,86,159,75,151)(68,87,160,76,152)(69,88,153,77,145)(70,81,154,78,146)(71,82,155,79,147)(72,83,156,80,148), (1,65,15,111)(2,112,16,66)(3,67,9,105)(4,106,10,68)(5,69,11,107)(6,108,12,70)(7,71,13,109)(8,110,14,72)(17,43,133,75)(18,76,134,44)(19,45,135,77)(20,78,136,46)(21,47,129,79)(22,80,130,48)(23,41,131,73)(24,74,132,42)(25,114,57,159)(26,160,58,115)(27,116,59,153)(28,154,60,117)(29,118,61,155)(30,156,62,119)(31,120,63,157)(32,158,64,113)(33,55,149,93)(34,94,150,56)(35,49,151,95)(36,96,152,50)(37,51,145,89)(38,90,146,52)(39,53,147,91)(40,92,148,54)(81,144,128,99)(82,100,121,137)(83,138,122,101)(84,102,123,139)(85,140,124,103)(86,104,125,141)(87,142,126,97)(88,98,127,143), (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), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,129)(18,134)(19,131)(20,136)(21,133)(22,130)(23,135)(24,132)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(32,64)(33,145)(34,150)(35,147)(36,152)(37,149)(38,146)(39,151)(40,148)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,91)(50,96)(51,93)(52,90)(53,95)(54,92)(55,89)(56,94)(65,107)(66,112)(67,109)(68,106)(69,111)(70,108)(71,105)(72,110)(81,128)(82,125)(83,122)(84,127)(85,124)(86,121)(87,126)(88,123)(97,142)(98,139)(99,144)(100,141)(101,138)(102,143)(103,140)(104,137)(113,158)(114,155)(115,160)(116,157)(117,154)(118,159)(119,156)(120,153) );
G=PermutationGroup([[(1,139,63,131,55),(2,140,64,132,56),(3,141,57,133,49),(4,142,58,134,50),(5,143,59,135,51),(6,144,60,136,52),(7,137,61,129,53),(8,138,62,130,54),(9,104,25,17,95),(10,97,26,18,96),(11,98,27,19,89),(12,99,28,20,90),(13,100,29,21,91),(14,101,30,22,92),(15,102,31,23,93),(16,103,32,24,94),(33,111,123,120,41),(34,112,124,113,42),(35,105,125,114,43),(36,106,126,115,44),(37,107,127,116,45),(38,108,128,117,46),(39,109,121,118,47),(40,110,122,119,48),(65,84,157,73,149),(66,85,158,74,150),(67,86,159,75,151),(68,87,160,76,152),(69,88,153,77,145),(70,81,154,78,146),(71,82,155,79,147),(72,83,156,80,148)], [(1,65,15,111),(2,112,16,66),(3,67,9,105),(4,106,10,68),(5,69,11,107),(6,108,12,70),(7,71,13,109),(8,110,14,72),(17,43,133,75),(18,76,134,44),(19,45,135,77),(20,78,136,46),(21,47,129,79),(22,80,130,48),(23,41,131,73),(24,74,132,42),(25,114,57,159),(26,160,58,115),(27,116,59,153),(28,154,60,117),(29,118,61,155),(30,156,62,119),(31,120,63,157),(32,158,64,113),(33,55,149,93),(34,94,150,56),(35,49,151,95),(36,96,152,50),(37,51,145,89),(38,90,146,52),(39,53,147,91),(40,92,148,54),(81,144,128,99),(82,100,121,137),(83,138,122,101),(84,102,123,139),(85,140,124,103),(86,104,125,141),(87,142,126,97),(88,98,127,143)], [(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)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,129),(18,134),(19,131),(20,136),(21,133),(22,130),(23,135),(24,132),(25,61),(26,58),(27,63),(28,60),(29,57),(30,62),(31,59),(32,64),(33,145),(34,150),(35,147),(36,152),(37,149),(38,146),(39,151),(40,148),(41,77),(42,74),(43,79),(44,76),(45,73),(46,78),(47,75),(48,80),(49,91),(50,96),(51,93),(52,90),(53,95),(54,92),(55,89),(56,94),(65,107),(66,112),(67,109),(68,106),(69,111),(70,108),(71,105),(72,110),(81,128),(82,125),(83,122),(84,127),(85,124),(86,121),(87,126),(88,123),(97,142),(98,139),(99,144),(100,141),(101,138),(102,143),(103,140),(104,137),(113,158),(114,155),(115,160),(116,157),(117,154),(118,159),(119,156),(120,153)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20BD | 40A | ··· | 40AF |
| order | 1 | 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 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | D4 | Q8 | M4(2) | C5×D4 | C5×Q8 | C5×M4(2) |
| kernel | C5×C4⋊M4(2) | C5×C4⋊C8 | C2×C4×C20 | C10×M4(2) | C4×C20 | C22×C20 | C4⋊M4(2) | C4⋊C8 | C2×C42 | C2×M4(2) | C42 | C22×C4 | C2×C20 | C2×C20 | C20 | C2×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 4 | 16 | 4 | 8 | 16 | 16 | 2 | 2 | 8 | 8 | 8 | 32 |
Matrix representation of C5×C4⋊M4(2) ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 39 | 1 |
| 0 | 39 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 26 | 24 |
| 0 | 0 | 18 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[9,0,0,0,0,32,0,0,0,0,40,39,0,0,1,1],[0,16,0,0,39,0,0,0,0,0,26,18,0,0,24,15],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40] >;
C5×C4⋊M4(2) in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes M_4(2)
% in TeX
G:=Group("C5xC4:M4(2)"); // GroupNames label
G:=SmallGroup(320,924);
// by ID
G=gap.SmallGroup(320,924);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,3446,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^5>;
// generators/relations