metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊9D4, C5⋊2C8⋊10D4, C8⋊6(C5⋊D4), C5⋊5(C8⋊3D4), C40⋊8C4⋊6C2, C4.26(D4×D5), (C2×D40)⋊25C2, C20⋊D4⋊6C2, (C2×SD16)⋊2D5, (C2×C8).91D10, (C10×SD16)⋊4C2, (C2×D4).77D10, C20.179(C2×D4), (C2×Q8).58D10, C20.23D4⋊5C2, (C2×Dic5).82D4, C22.273(D4×D5), C2.23(C20⋊D4), C10.32(C4⋊1D4), C2.31(D40⋊C2), C10.81(C8⋊C22), (C2×C40).116C22, (C2×C20).453C23, (Q8×C10).82C22, (C2×D20).127C22, (D4×C10).102C22, (C4×Dic5).58C22, (C2×D4⋊D5)⋊21C2, (C2×Q8⋊D5)⋊19C2, C4.10(C2×C5⋊D4), (C2×C10).365(C2×D4), (C2×C4).542(C22×D5), (C2×C5⋊2C8).161C22, SmallGroup(320,803)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊9D4
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a-1, cbc=b-1 >
Subgroups: 750 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C2×SD16, C5⋊2C8, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C8⋊3D4, D40, C2×C5⋊2C8, C4×Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C2×C40, C5×SD16, C2×D20, C2×C5⋊D4, D4×C10, Q8×C10, C40⋊8C4, C2×D40, C2×D4⋊D5, C20⋊D4, C2×Q8⋊D5, C20.23D4, C10×SD16, C40⋊9D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8⋊C22, C5⋊D4, C22×D5, C8⋊3D4, D4×D5, C2×C5⋊D4, D40⋊C2, C20⋊D4, C40⋊9D4
►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)
(1 102 146 73)(2 91 147 62)(3 120 148 51)(4 109 149 80)(5 98 150 69)(6 87 151 58)(7 116 152 47)(8 105 153 76)(9 94 154 65)(10 83 155 54)(11 112 156 43)(12 101 157 72)(13 90 158 61)(14 119 159 50)(15 108 160 79)(16 97 121 68)(17 86 122 57)(18 115 123 46)(19 104 124 75)(20 93 125 64)(21 82 126 53)(22 111 127 42)(23 100 128 71)(24 89 129 60)(25 118 130 49)(26 107 131 78)(27 96 132 67)(28 85 133 56)(29 114 134 45)(30 103 135 74)(31 92 136 63)(32 81 137 52)(33 110 138 41)(34 99 139 70)(35 88 140 59)(36 117 141 48)(37 106 142 77)(38 95 143 66)(39 84 144 55)(40 113 145 44)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(121 131)(122 130)(123 129)(124 128)(125 127)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)(141 151)(142 150)(143 149)(144 148)(145 147)
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), (1,102,146,73)(2,91,147,62)(3,120,148,51)(4,109,149,80)(5,98,150,69)(6,87,151,58)(7,116,152,47)(8,105,153,76)(9,94,154,65)(10,83,155,54)(11,112,156,43)(12,101,157,72)(13,90,158,61)(14,119,159,50)(15,108,160,79)(16,97,121,68)(17,86,122,57)(18,115,123,46)(19,104,124,75)(20,93,125,64)(21,82,126,53)(22,111,127,42)(23,100,128,71)(24,89,129,60)(25,118,130,49)(26,107,131,78)(27,96,132,67)(28,85,133,56)(29,114,134,45)(30,103,135,74)(31,92,136,63)(32,81,137,52)(33,110,138,41)(34,99,139,70)(35,88,140,59)(36,117,141,48)(37,106,142,77)(38,95,143,66)(39,84,144,55)(40,113,145,44), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(121,131)(122,130)(123,129)(124,128)(125,127)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)>;
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), (1,102,146,73)(2,91,147,62)(3,120,148,51)(4,109,149,80)(5,98,150,69)(6,87,151,58)(7,116,152,47)(8,105,153,76)(9,94,154,65)(10,83,155,54)(11,112,156,43)(12,101,157,72)(13,90,158,61)(14,119,159,50)(15,108,160,79)(16,97,121,68)(17,86,122,57)(18,115,123,46)(19,104,124,75)(20,93,125,64)(21,82,126,53)(22,111,127,42)(23,100,128,71)(24,89,129,60)(25,118,130,49)(26,107,131,78)(27,96,132,67)(28,85,133,56)(29,114,134,45)(30,103,135,74)(31,92,136,63)(32,81,137,52)(33,110,138,41)(34,99,139,70)(35,88,140,59)(36,117,141,48)(37,106,142,77)(38,95,143,66)(39,84,144,55)(40,113,145,44), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(121,131)(122,130)(123,129)(124,128)(125,127)(132,160)(133,159)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147) );
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)], [(1,102,146,73),(2,91,147,62),(3,120,148,51),(4,109,149,80),(5,98,150,69),(6,87,151,58),(7,116,152,47),(8,105,153,76),(9,94,154,65),(10,83,155,54),(11,112,156,43),(12,101,157,72),(13,90,158,61),(14,119,159,50),(15,108,160,79),(16,97,121,68),(17,86,122,57),(18,115,123,46),(19,104,124,75),(20,93,125,64),(21,82,126,53),(22,111,127,42),(23,100,128,71),(24,89,129,60),(25,118,130,49),(26,107,131,78),(27,96,132,67),(28,85,133,56),(29,114,134,45),(30,103,135,74),(31,92,136,63),(32,81,137,52),(33,110,138,41),(34,99,139,70),(35,88,140,59),(36,117,141,48),(37,106,142,77),(38,95,143,66),(39,84,144,55),(40,113,145,44)], [(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,120),(56,119),(57,118),(58,117),(59,116),(60,115),(61,114),(62,113),(63,112),(64,111),(65,110),(66,109),(67,108),(68,107),(69,106),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,96),(80,95),(121,131),(122,130),(123,129),(124,128),(125,127),(132,160),(133,159),(134,158),(135,157),(136,156),(137,155),(138,154),(139,153),(140,152),(141,151),(142,150),(143,149),(144,148),(145,147)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 40 | 40 | 2 | 2 | 8 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D40⋊C2 |
| kernel | C40⋊9D4 | C40⋊8C4 | C2×D40 | C2×D4⋊D5 | C20⋊D4 | C2×Q8⋊D5 | C20.23D4 | C10×SD16 | C5⋊2C8 | C40 | C2×Dic5 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40⋊9D4 ►in GL8(𝔽41)
| 28 | 19 | 25 | 14 | 0 | 0 | 0 | 0 |
| 22 | 19 | 27 | 27 | 0 | 0 | 0 | 0 |
| 16 | 27 | 13 | 22 | 0 | 0 | 0 | 0 |
| 14 | 14 | 19 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 4 | 32 | 4 |
| 0 | 0 | 0 | 0 | 37 | 4 | 37 | 4 |
| 0 | 0 | 0 | 0 | 9 | 37 | 32 | 4 |
| 0 | 0 | 0 | 0 | 4 | 37 | 37 | 4 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 | 15 | 16 |
| 0 | 0 | 0 | 0 | 25 | 27 | 34 | 26 |
| 0 | 0 | 0 | 0 | 26 | 25 | 14 | 0 |
| 0 | 0 | 0 | 0 | 7 | 15 | 25 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 1 |
G:=sub<GL(8,GF(41))| [28,22,16,14,0,0,0,0,19,19,27,14,0,0,0,0,25,27,13,19,0,0,0,0,14,27,22,22,0,0,0,0,0,0,0,0,32,37,9,4,0,0,0,0,4,4,37,37,0,0,0,0,32,37,32,37,0,0,0,0,4,4,4,4],[0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,14,25,26,7,0,0,0,0,0,27,25,15,0,0,0,0,15,34,14,25,0,0,0,0,16,26,0,27],[1,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,1] >;
C40⋊9D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_9D_4 % in TeX
G:=Group("C40:9D4"); // GroupNames label
G:=SmallGroup(320,803);
// by ID
G=gap.SmallGroup(320,803);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations