metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊18D4, Dic5⋊3M4(2), C8⋊9(C5⋊D4), C5⋊8(C8⋊6D4), (C8×Dic5)⋊31C2, C10.110(C4×D4), C20.443(C2×D4), (C2×C8).276D10, (C2×M4(2))⋊8D5, D10⋊1C8⋊39C2, C23.20(C4×D5), C10.57(C8○D4), (C10×M4(2))⋊7C2, C20.8Q8⋊41C2, C2.22(D5×M4(2)), C23.D5.20C4, D10⋊C4.27C4, C4.138(C4○D20), C20.254(C4○D4), C20.55D4⋊30C2, (C2×C20).868C23, (C2×C40).318C22, C10.D4.27C4, (C22×C4).137D10, C10.67(C2×M4(2)), C2.18(D20.2C4), (C22×C20).375C22, (C4×Dic5).317C22, (C2×C4).51(C4×D5), C2.25(C4×C5⋊D4), (C2×C8⋊D5)⋊25C2, (C2×C5⋊D4).20C4, (C4×C5⋊D4).17C2, C4.134(C2×C5⋊D4), C22.147(C2×C4×D5), (C2×C20).361(C2×C4), (C2×C4×D5).236C22, (C22×D5).30(C2×C4), (C2×C4).810(C22×D5), (C22×C10).136(C2×C4), (C2×C10).239(C22×C4), (C2×C5⋊2C8).332C22, (C2×Dic5).113(C2×C4), SmallGroup(320,755)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊18D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a29, cbc=b-1 >
Subgroups: 382 in 122 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C2×M4(2), C5⋊2C8, C40, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊6D4, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C5×M4(2), C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D10⋊1C8, C20.55D4, C2×C8⋊D5, C4×C5⋊D4, C10×M4(2), C40⋊18D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C5⋊D4, C22×D5, C8⋊6D4, C2×C4×D5, C4○D20, C2×C5⋊D4, D5×M4(2), D20.2C4, C4×C5⋊D4, C40⋊18D4
►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 71 121 98)(2 80 122 107)(3 49 123 116)(4 58 124 85)(5 67 125 94)(6 76 126 103)(7 45 127 112)(8 54 128 81)(9 63 129 90)(10 72 130 99)(11 41 131 108)(12 50 132 117)(13 59 133 86)(14 68 134 95)(15 77 135 104)(16 46 136 113)(17 55 137 82)(18 64 138 91)(19 73 139 100)(20 42 140 109)(21 51 141 118)(22 60 142 87)(23 69 143 96)(24 78 144 105)(25 47 145 114)(26 56 146 83)(27 65 147 92)(28 74 148 101)(29 43 149 110)(30 52 150 119)(31 61 151 88)(32 70 152 97)(33 79 153 106)(34 48 154 115)(35 57 155 84)(36 66 156 93)(37 75 157 102)(38 44 158 111)(39 53 159 120)(40 62 160 89)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 108)(42 97)(43 86)(44 115)(45 104)(46 93)(47 82)(48 111)(49 100)(50 89)(51 118)(52 107)(53 96)(54 85)(55 114)(56 103)(57 92)(58 81)(59 110)(60 99)(61 88)(62 117)(63 106)(64 95)(65 84)(66 113)(67 102)(68 91)(69 120)(70 109)(71 98)(72 87)(73 116)(74 105)(75 94)(76 83)(77 112)(78 101)(79 90)(80 119)(122 150)(123 139)(124 128)(125 157)(126 146)(127 135)(129 153)(130 142)(132 160)(133 149)(134 138)(136 156)(137 145)(140 152)(143 159)(144 148)(147 155)(154 158)
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), (1,71,121,98)(2,80,122,107)(3,49,123,116)(4,58,124,85)(5,67,125,94)(6,76,126,103)(7,45,127,112)(8,54,128,81)(9,63,129,90)(10,72,130,99)(11,41,131,108)(12,50,132,117)(13,59,133,86)(14,68,134,95)(15,77,135,104)(16,46,136,113)(17,55,137,82)(18,64,138,91)(19,73,139,100)(20,42,140,109)(21,51,141,118)(22,60,142,87)(23,69,143,96)(24,78,144,105)(25,47,145,114)(26,56,146,83)(27,65,147,92)(28,74,148,101)(29,43,149,110)(30,52,150,119)(31,61,151,88)(32,70,152,97)(33,79,153,106)(34,48,154,115)(35,57,155,84)(36,66,156,93)(37,75,157,102)(38,44,158,111)(39,53,159,120)(40,62,160,89), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,108)(42,97)(43,86)(44,115)(45,104)(46,93)(47,82)(48,111)(49,100)(50,89)(51,118)(52,107)(53,96)(54,85)(55,114)(56,103)(57,92)(58,81)(59,110)(60,99)(61,88)(62,117)(63,106)(64,95)(65,84)(66,113)(67,102)(68,91)(69,120)(70,109)(71,98)(72,87)(73,116)(74,105)(75,94)(76,83)(77,112)(78,101)(79,90)(80,119)(122,150)(123,139)(124,128)(125,157)(126,146)(127,135)(129,153)(130,142)(132,160)(133,149)(134,138)(136,156)(137,145)(140,152)(143,159)(144,148)(147,155)(154,158)>;
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), (1,71,121,98)(2,80,122,107)(3,49,123,116)(4,58,124,85)(5,67,125,94)(6,76,126,103)(7,45,127,112)(8,54,128,81)(9,63,129,90)(10,72,130,99)(11,41,131,108)(12,50,132,117)(13,59,133,86)(14,68,134,95)(15,77,135,104)(16,46,136,113)(17,55,137,82)(18,64,138,91)(19,73,139,100)(20,42,140,109)(21,51,141,118)(22,60,142,87)(23,69,143,96)(24,78,144,105)(25,47,145,114)(26,56,146,83)(27,65,147,92)(28,74,148,101)(29,43,149,110)(30,52,150,119)(31,61,151,88)(32,70,152,97)(33,79,153,106)(34,48,154,115)(35,57,155,84)(36,66,156,93)(37,75,157,102)(38,44,158,111)(39,53,159,120)(40,62,160,89), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,108)(42,97)(43,86)(44,115)(45,104)(46,93)(47,82)(48,111)(49,100)(50,89)(51,118)(52,107)(53,96)(54,85)(55,114)(56,103)(57,92)(58,81)(59,110)(60,99)(61,88)(62,117)(63,106)(64,95)(65,84)(66,113)(67,102)(68,91)(69,120)(70,109)(71,98)(72,87)(73,116)(74,105)(75,94)(76,83)(77,112)(78,101)(79,90)(80,119)(122,150)(123,139)(124,128)(125,157)(126,146)(127,135)(129,153)(130,142)(132,160)(133,149)(134,138)(136,156)(137,145)(140,152)(143,159)(144,148)(147,155)(154,158) );
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)], [(1,71,121,98),(2,80,122,107),(3,49,123,116),(4,58,124,85),(5,67,125,94),(6,76,126,103),(7,45,127,112),(8,54,128,81),(9,63,129,90),(10,72,130,99),(11,41,131,108),(12,50,132,117),(13,59,133,86),(14,68,134,95),(15,77,135,104),(16,46,136,113),(17,55,137,82),(18,64,138,91),(19,73,139,100),(20,42,140,109),(21,51,141,118),(22,60,142,87),(23,69,143,96),(24,78,144,105),(25,47,145,114),(26,56,146,83),(27,65,147,92),(28,74,148,101),(29,43,149,110),(30,52,150,119),(31,61,151,88),(32,70,152,97),(33,79,153,106),(34,48,154,115),(35,57,155,84),(36,66,156,93),(37,75,157,102),(38,44,158,111),(39,53,159,120),(40,62,160,89)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,108),(42,97),(43,86),(44,115),(45,104),(46,93),(47,82),(48,111),(49,100),(50,89),(51,118),(52,107),(53,96),(54,85),(55,114),(56,103),(57,92),(58,81),(59,110),(60,99),(61,88),(62,117),(63,106),(64,95),(65,84),(66,113),(67,102),(68,91),(69,120),(70,109),(71,98),(72,87),(73,116),(74,105),(75,94),(76,83),(77,112),(78,101),(79,90),(80,119),(122,150),(123,139),(124,128),(125,157),(126,146),(127,135),(129,153),(130,142),(132,160),(133,149),(134,138),(136,156),(137,145),(140,152),(143,159),(144,148),(147,155),(154,158)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | C5⋊D4 | C4×D5 | C4×D5 | C4○D20 | D5×M4(2) | D20.2C4 |
| kernel | C40⋊18D4 | C8×Dic5 | C20.8Q8 | D10⋊1C8 | C20.55D4 | C2×C8⋊D5 | C4×C5⋊D4 | C10×M4(2) | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C40 | C2×M4(2) | Dic5 | C20 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 4 | 4 |
Matrix representation of C40⋊18D4 ►in GL4(𝔽41) generated by
| 40 | 34 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 32 | 0 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,7,0,0,34,7,0,0,0,0,0,32,0,0,1,0],[17,3,0,0,40,24,0,0,0,0,40,0,0,0,0,40],[1,34,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C40⋊18D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{18}D_4 % in TeX
G:=Group("C40:18D4"); // GroupNames label
G:=SmallGroup(320,755);
// by ID
G=gap.SmallGroup(320,755);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations