direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C40⋊C2, C20⋊8SD16, C42.258D10, (C4×C8)⋊11D5, C8⋊11(C4×D5), C5⋊2(C4×SD16), (C4×C40)⋊16C2, C40⋊33(C2×C4), C2.9(C4×D20), C40⋊6C4⋊28C2, (C4×D20).3C2, C10.36(C4×D4), (C2×C4).60D20, (C4×Dic10)⋊1C2, D20.26(C2×C4), C10.2(C4○D8), (C2×C8).285D10, (C2×C20).350D4, C10.3(C2×SD16), Dic10⋊17(C2×C4), C22.27(C2×D20), D20⋊5C4.17C2, C20.216(C4○D4), C4.100(C4○D20), C2.1(D40⋊7C2), C20.44D4⋊43C2, (C2×C40).345C22, (C4×C20).325C22, C20.160(C22×C4), (C2×C20).720C23, (C2×D20).193C22, C4⋊Dic5.262C22, (C2×Dic10).212C22, C4.59(C2×C4×D5), C2.1(C2×C40⋊C2), (C2×C40⋊C2).10C2, (C2×C10).103(C2×D4), (C2×C4).663(C22×D5), SmallGroup(320,318)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C40⋊C2
G = < a,b,c | a4=b40=c2=1, ab=ba, ac=ca, cbc=b19 >
Subgroups: 518 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C40, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, C4×SD16, C40⋊C2, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, C40⋊6C4, D20⋊5C4, C4×C40, C4×Dic10, C4×D20, C2×C40⋊C2, C4×C40⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×SD16, C4○D8, C4×D5, D20, C22×D5, C4×SD16, C40⋊C2, C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×C40⋊C2, D40⋊7C2, C4×C40⋊C2
►On 160 points
Generators in S160
(1 126 99 58)(2 127 100 59)(3 128 101 60)(4 129 102 61)(5 130 103 62)(6 131 104 63)(7 132 105 64)(8 133 106 65)(9 134 107 66)(10 135 108 67)(11 136 109 68)(12 137 110 69)(13 138 111 70)(14 139 112 71)(15 140 113 72)(16 141 114 73)(17 142 115 74)(18 143 116 75)(19 144 117 76)(20 145 118 77)(21 146 119 78)(22 147 120 79)(23 148 81 80)(24 149 82 41)(25 150 83 42)(26 151 84 43)(27 152 85 44)(28 153 86 45)(29 154 87 46)(30 155 88 47)(31 156 89 48)(32 157 90 49)(33 158 91 50)(34 159 92 51)(35 160 93 52)(36 121 94 53)(37 122 95 54)(38 123 96 55)(39 124 97 56)(40 125 98 57)
(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 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 55)(42 74)(43 53)(44 72)(45 51)(46 70)(47 49)(48 68)(50 66)(52 64)(54 62)(56 60)(57 79)(59 77)(61 75)(63 73)(65 71)(67 69)(76 80)(81 117)(82 96)(83 115)(84 94)(85 113)(86 92)(87 111)(88 90)(89 109)(91 107)(93 105)(95 103)(97 101)(98 120)(100 118)(102 116)(104 114)(106 112)(108 110)(121 151)(122 130)(123 149)(124 128)(125 147)(127 145)(129 143)(131 141)(132 160)(133 139)(134 158)(135 137)(136 156)(138 154)(140 152)(142 150)(144 148)(153 159)(155 157)
G:=sub<Sym(160)| (1,126,99,58)(2,127,100,59)(3,128,101,60)(4,129,102,61)(5,130,103,62)(6,131,104,63)(7,132,105,64)(8,133,106,65)(9,134,107,66)(10,135,108,67)(11,136,109,68)(12,137,110,69)(13,138,111,70)(14,139,112,71)(15,140,113,72)(16,141,114,73)(17,142,115,74)(18,143,116,75)(19,144,117,76)(20,145,118,77)(21,146,119,78)(22,147,120,79)(23,148,81,80)(24,149,82,41)(25,150,83,42)(26,151,84,43)(27,152,85,44)(28,153,86,45)(29,154,87,46)(30,155,88,47)(31,156,89,48)(32,157,90,49)(33,158,91,50)(34,159,92,51)(35,160,93,52)(36,121,94,53)(37,122,95,54)(38,123,96,55)(39,124,97,56)(40,125,98,57), (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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,55)(42,74)(43,53)(44,72)(45,51)(46,70)(47,49)(48,68)(50,66)(52,64)(54,62)(56,60)(57,79)(59,77)(61,75)(63,73)(65,71)(67,69)(76,80)(81,117)(82,96)(83,115)(84,94)(85,113)(86,92)(87,111)(88,90)(89,109)(91,107)(93,105)(95,103)(97,101)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110)(121,151)(122,130)(123,149)(124,128)(125,147)(127,145)(129,143)(131,141)(132,160)(133,139)(134,158)(135,137)(136,156)(138,154)(140,152)(142,150)(144,148)(153,159)(155,157)>;
G:=Group( (1,126,99,58)(2,127,100,59)(3,128,101,60)(4,129,102,61)(5,130,103,62)(6,131,104,63)(7,132,105,64)(8,133,106,65)(9,134,107,66)(10,135,108,67)(11,136,109,68)(12,137,110,69)(13,138,111,70)(14,139,112,71)(15,140,113,72)(16,141,114,73)(17,142,115,74)(18,143,116,75)(19,144,117,76)(20,145,118,77)(21,146,119,78)(22,147,120,79)(23,148,81,80)(24,149,82,41)(25,150,83,42)(26,151,84,43)(27,152,85,44)(28,153,86,45)(29,154,87,46)(30,155,88,47)(31,156,89,48)(32,157,90,49)(33,158,91,50)(34,159,92,51)(35,160,93,52)(36,121,94,53)(37,122,95,54)(38,123,96,55)(39,124,97,56)(40,125,98,57), (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), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,55)(42,74)(43,53)(44,72)(45,51)(46,70)(47,49)(48,68)(50,66)(52,64)(54,62)(56,60)(57,79)(59,77)(61,75)(63,73)(65,71)(67,69)(76,80)(81,117)(82,96)(83,115)(84,94)(85,113)(86,92)(87,111)(88,90)(89,109)(91,107)(93,105)(95,103)(97,101)(98,120)(100,118)(102,116)(104,114)(106,112)(108,110)(121,151)(122,130)(123,149)(124,128)(125,147)(127,145)(129,143)(131,141)(132,160)(133,139)(134,158)(135,137)(136,156)(138,154)(140,152)(142,150)(144,148)(153,159)(155,157) );
G=PermutationGroup([[(1,126,99,58),(2,127,100,59),(3,128,101,60),(4,129,102,61),(5,130,103,62),(6,131,104,63),(7,132,105,64),(8,133,106,65),(9,134,107,66),(10,135,108,67),(11,136,109,68),(12,137,110,69),(13,138,111,70),(14,139,112,71),(15,140,113,72),(16,141,114,73),(17,142,115,74),(18,143,116,75),(19,144,117,76),(20,145,118,77),(21,146,119,78),(22,147,120,79),(23,148,81,80),(24,149,82,41),(25,150,83,42),(26,151,84,43),(27,152,85,44),(28,153,86,45),(29,154,87,46),(30,155,88,47),(31,156,89,48),(32,157,90,49),(33,158,91,50),(34,159,92,51),(35,160,93,52),(36,121,94,53),(37,122,95,54),(38,123,96,55),(39,124,97,56),(40,125,98,57)], [(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)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,55),(42,74),(43,53),(44,72),(45,51),(46,70),(47,49),(48,68),(50,66),(52,64),(54,62),(56,60),(57,79),(59,77),(61,75),(63,73),(65,71),(67,69),(76,80),(81,117),(82,96),(83,115),(84,94),(85,113),(86,92),(87,111),(88,90),(89,109),(91,107),(93,105),(95,103),(97,101),(98,120),(100,118),(102,116),(104,114),(106,112),(108,110),(121,151),(122,130),(123,149),(124,128),(125,147),(127,145),(129,143),(131,141),(132,160),(133,139),(134,158),(135,137),(136,156),(138,154),(140,152),(142,150),(144,148),(153,159),(155,157)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | C4○D8 | C4×D5 | D20 | C40⋊C2 | C4○D20 | D40⋊7C2 |
| kernel | C4×C40⋊C2 | C20.44D4 | C40⋊6C4 | D20⋊5C4 | C4×C40 | C4×Dic10 | C4×D20 | C2×C40⋊C2 | C40⋊C2 | C2×C20 | C4×C8 | C20 | C20 | C42 | C2×C8 | C10 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 8 | 16 |
Matrix representation of C4×C40⋊C2 ►in GL4(𝔽41) generated by
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 6 | 6 | 0 | 0 |
| 35 | 1 | 0 | 0 |
| 0 | 0 | 39 | 23 |
| 0 | 0 | 33 | 31 |
| 1 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 6 | 7 |
| 0 | 0 | 36 | 35 |
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[6,35,0,0,6,1,0,0,0,0,39,33,0,0,23,31],[1,6,0,0,0,40,0,0,0,0,6,36,0,0,7,35] >;
C4×C40⋊C2 in GAP, Magma, Sage, TeX
C_4\times C_{40}\rtimes C_2 % in TeX
G:=Group("C4xC40:C2"); // GroupNames label
G:=SmallGroup(320,318);
// by ID
G=gap.SmallGroup(320,318);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,58,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^4=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations