direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42.7C22, (C4×C8)⋊3C10, (C4×C40)⋊8C2, C4⋊C8⋊14C10, C4⋊C4.6C20, C8⋊C4⋊8C10, C22⋊C4.3C20, C22⋊C8.8C10, C10.71(C8○D4), C23.11(C2×C20), C42.61(C2×C10), C20.352(C4○D4), (C2×C40).327C22, (C4×C20).247C22, (C2×C20).989C23, C42⋊C2.8C10, C22.46(C22×C20), C10.80(C42⋊C2), (C22×C20).416C22, (C5×C4⋊C8)⋊33C2, C2.6(C5×C8○D4), (C5×C4⋊C4).31C4, (C5×C8⋊C4)⋊22C2, C4.50(C5×C4○D4), (C2×C4).27(C2×C20), (C2×C8).51(C2×C10), (C2×C20).373(C2×C4), (C5×C22⋊C4).18C4, (C5×C22⋊C8).17C2, (C22×C4).35(C2×C10), (C22×C10).90(C2×C4), C2.12(C5×C42⋊C2), (C2×C10).340(C22×C4), (C5×C42⋊C2).22C2, (C2×C4).157(C22×C10), SmallGroup(320,934)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42.7C22
G = < a,b,c,d,e | a5=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe=bc2, cd=dc, ce=ec, ede=b2c2d >
Subgroups: 130 in 96 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C40, C2×C20, C2×C20, C2×C20, C22×C10, C42.7C22, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C4×C40, C5×C8⋊C4, C5×C22⋊C8, C5×C4⋊C8, C5×C42⋊C2, C5×C42.7C22
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C4○D4, C20, C2×C10, C42⋊C2, C8○D4, C2×C20, C22×C10, C42.7C22, C22×C20, C5×C4○D4, C5×C42⋊C2, C5×C8○D4, C5×C42.7C22
►On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 42 114 34 94)(10 43 115 35 95)(11 44 116 36 96)(12 45 117 37 89)(13 46 118 38 90)(14 47 119 39 91)(15 48 120 40 92)(16 41 113 33 93)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 85 153 73 145)(66 86 154 74 146)(67 87 155 75 147)(68 88 156 76 148)(69 81 157 77 149)(70 82 158 78 150)(71 83 159 79 151)(72 84 160 80 152)
(1 71 123 37)(2 34 124 68)(3 65 125 39)(4 36 126 70)(5 67 127 33)(6 38 128 72)(7 69 121 35)(8 40 122 66)(9 60 156 26)(10 31 157 57)(11 62 158 28)(12 25 159 59)(13 64 160 30)(14 27 153 61)(15 58 154 32)(16 29 155 63)(17 151 51 117)(18 114 52 148)(19 145 53 119)(20 116 54 150)(21 147 55 113)(22 118 56 152)(23 149 49 115)(24 120 50 146)(41 101 75 135)(42 132 76 98)(43 103 77 129)(44 134 78 100)(45 97 79 131)(46 136 80 102)(47 99 73 133)(48 130 74 104)(81 137 95 111)(82 108 96 142)(83 139 89 105)(84 110 90 144)(85 141 91 107)(86 112 92 138)(87 143 93 109)(88 106 94 140)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(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)
(2 128)(4 122)(6 124)(8 126)(9 156)(10 14)(11 158)(12 16)(13 160)(15 154)(18 56)(20 50)(22 52)(24 54)(26 64)(28 58)(30 60)(32 62)(33 37)(34 68)(35 39)(36 70)(38 72)(40 66)(41 45)(42 76)(43 47)(44 78)(46 80)(48 74)(65 69)(67 71)(73 77)(75 79)(81 85)(82 96)(83 87)(84 90)(86 92)(88 94)(89 93)(91 95)(98 136)(100 130)(102 132)(104 134)(106 144)(108 138)(110 140)(112 142)(113 117)(114 148)(115 119)(116 150)(118 152)(120 146)(145 149)(147 151)(153 157)(155 159)
G:=sub<Sym(160)| (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,42,114,34,94)(10,43,115,35,95)(11,44,116,36,96)(12,45,117,37,89)(13,46,118,38,90)(14,47,119,39,91)(15,48,120,40,92)(16,41,113,33,93)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,71,123,37)(2,34,124,68)(3,65,125,39)(4,36,126,70)(5,67,127,33)(6,38,128,72)(7,69,121,35)(8,40,122,66)(9,60,156,26)(10,31,157,57)(11,62,158,28)(12,25,159,59)(13,64,160,30)(14,27,153,61)(15,58,154,32)(16,29,155,63)(17,151,51,117)(18,114,52,148)(19,145,53,119)(20,116,54,150)(21,147,55,113)(22,118,56,152)(23,149,49,115)(24,120,50,146)(41,101,75,135)(42,132,76,98)(43,103,77,129)(44,134,78,100)(45,97,79,131)(46,136,80,102)(47,99,73,133)(48,130,74,104)(81,137,95,111)(82,108,96,142)(83,139,89,105)(84,110,90,144)(85,141,91,107)(86,112,92,138)(87,143,93,109)(88,106,94,140), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160), (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), (2,128)(4,122)(6,124)(8,126)(9,156)(10,14)(11,158)(12,16)(13,160)(15,154)(18,56)(20,50)(22,52)(24,54)(26,64)(28,58)(30,60)(32,62)(33,37)(34,68)(35,39)(36,70)(38,72)(40,66)(41,45)(42,76)(43,47)(44,78)(46,80)(48,74)(65,69)(67,71)(73,77)(75,79)(81,85)(82,96)(83,87)(84,90)(86,92)(88,94)(89,93)(91,95)(98,136)(100,130)(102,132)(104,134)(106,144)(108,138)(110,140)(112,142)(113,117)(114,148)(115,119)(116,150)(118,152)(120,146)(145,149)(147,151)(153,157)(155,159)>;
G:=Group( (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,42,114,34,94)(10,43,115,35,95)(11,44,116,36,96)(12,45,117,37,89)(13,46,118,38,90)(14,47,119,39,91)(15,48,120,40,92)(16,41,113,33,93)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,71,123,37)(2,34,124,68)(3,65,125,39)(4,36,126,70)(5,67,127,33)(6,38,128,72)(7,69,121,35)(8,40,122,66)(9,60,156,26)(10,31,157,57)(11,62,158,28)(12,25,159,59)(13,64,160,30)(14,27,153,61)(15,58,154,32)(16,29,155,63)(17,151,51,117)(18,114,52,148)(19,145,53,119)(20,116,54,150)(21,147,55,113)(22,118,56,152)(23,149,49,115)(24,120,50,146)(41,101,75,135)(42,132,76,98)(43,103,77,129)(44,134,78,100)(45,97,79,131)(46,136,80,102)(47,99,73,133)(48,130,74,104)(81,137,95,111)(82,108,96,142)(83,139,89,105)(84,110,90,144)(85,141,91,107)(86,112,92,138)(87,143,93,109)(88,106,94,140), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160), (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), (2,128)(4,122)(6,124)(8,126)(9,156)(10,14)(11,158)(12,16)(13,160)(15,154)(18,56)(20,50)(22,52)(24,54)(26,64)(28,58)(30,60)(32,62)(33,37)(34,68)(35,39)(36,70)(38,72)(40,66)(41,45)(42,76)(43,47)(44,78)(46,80)(48,74)(65,69)(67,71)(73,77)(75,79)(81,85)(82,96)(83,87)(84,90)(86,92)(88,94)(89,93)(91,95)(98,136)(100,130)(102,132)(104,134)(106,144)(108,138)(110,140)(112,142)(113,117)(114,148)(115,119)(116,150)(118,152)(120,146)(145,149)(147,151)(153,157)(155,159) );
G=PermutationGroup([[(1,105,25,97,17),(2,106,26,98,18),(3,107,27,99,19),(4,108,28,100,20),(5,109,29,101,21),(6,110,30,102,22),(7,111,31,103,23),(8,112,32,104,24),(9,42,114,34,94),(10,43,115,35,95),(11,44,116,36,96),(12,45,117,37,89),(13,46,118,38,90),(14,47,119,39,91),(15,48,120,40,92),(16,41,113,33,93),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,85,153,73,145),(66,86,154,74,146),(67,87,155,75,147),(68,88,156,76,148),(69,81,157,77,149),(70,82,158,78,150),(71,83,159,79,151),(72,84,160,80,152)], [(1,71,123,37),(2,34,124,68),(3,65,125,39),(4,36,126,70),(5,67,127,33),(6,38,128,72),(7,69,121,35),(8,40,122,66),(9,60,156,26),(10,31,157,57),(11,62,158,28),(12,25,159,59),(13,64,160,30),(14,27,153,61),(15,58,154,32),(16,29,155,63),(17,151,51,117),(18,114,52,148),(19,145,53,119),(20,116,54,150),(21,147,55,113),(22,118,56,152),(23,149,49,115),(24,120,50,146),(41,101,75,135),(42,132,76,98),(43,103,77,129),(44,134,78,100),(45,97,79,131),(46,136,80,102),(47,99,73,133),(48,130,74,104),(81,137,95,111),(82,108,96,142),(83,139,89,105),(84,110,90,144),(85,141,91,107),(86,112,92,138),(87,143,93,109),(88,106,94,140)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160)], [(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)], [(2,128),(4,122),(6,124),(8,126),(9,156),(10,14),(11,158),(12,16),(13,160),(15,154),(18,56),(20,50),(22,52),(24,54),(26,64),(28,58),(30,60),(32,62),(33,37),(34,68),(35,39),(36,70),(38,72),(40,66),(41,45),(42,76),(43,47),(44,78),(46,80),(48,74),(65,69),(67,71),(73,77),(75,79),(81,85),(82,96),(83,87),(84,90),(86,92),(88,94),(89,93),(91,95),(98,136),(100,130),(102,132),(104,134),(106,144),(108,138),(110,140),(112,142),(113,117),(114,148),(115,119),(116,150),(118,152),(120,146),(145,149),(147,151),(153,157),(155,159)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AR | 40A | ··· | 40AF | 40AG | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C4○D4 | C8○D4 | C5×C4○D4 | C5×C8○D4 |
| kernel | C5×C42.7C22 | C4×C40 | C5×C8⋊C4 | C5×C22⋊C8 | C5×C4⋊C8 | C5×C42⋊C2 | C5×C22⋊C4 | C5×C4⋊C4 | C42.7C22 | C4×C8 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C42⋊C2 | C22⋊C4 | C4⋊C4 | C20 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 16 | 4 | 8 | 16 | 32 |
Matrix representation of C5×C42.7C22 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 2 | 0 | 0 |
| 1 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 26 | 9 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 38 | 13 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 29 | 39 |
| 0 | 0 | 10 | 12 |
| 1 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 29 | 40 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[9,1,0,0,2,32,0,0,0,0,32,26,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,13,3,0,0,0,0,29,10,0,0,39,12],[1,32,0,0,0,40,0,0,0,0,1,29,0,0,0,40] >;
C5×C42.7C22 in GAP, Magma, Sage, TeX
C_5\times C_4^2._7C_2^2
% in TeX
G:=Group("C5xC4^2.7C2^2"); // GroupNames label
G:=SmallGroup(320,934);
// by ID
G=gap.SmallGroup(320,934);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1731,226,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=b^2*c^2*d>;
// generators/relations