metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.37D4, (C2×Q16)⋊9D5, C4.28(D4×D5), C40⋊8C4⋊10C2, (C2×C8).96D10, C5⋊2C8.10D4, C8.6(C5⋊D4), C5⋊5(C8.2D4), (C10×Q16)⋊10C2, C20.188(C2×D4), (C2×Q8).66D10, Dic5⋊Q8⋊6C2, (C2×Dic5).86D4, C22.281(D4×D5), C2.25(C20⋊D4), C10.34(C4⋊1D4), (C2×C40).151C22, (C2×C20).464C23, C20.23D4.8C2, (Q8×C10).93C22, C2.31(Q16⋊D5), (C2×D20).130C22, C10.81(C8.C22), (C4×Dic5).62C22, (C2×Dic10).137C22, C4.15(C2×C5⋊D4), (C2×Q8⋊D5).9C2, (C2×C5⋊Q16)⋊21C2, (C2×C40⋊C2).8C2, (C2×C10).375(C2×D4), (C2×C4).552(C22×D5), (C2×C5⋊2C8).168C22, SmallGroup(320,817)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.37D4
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a19, cbc=a20b-1 >
Subgroups: 526 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×6], C23, D5, C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], Q16 [×4], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16, C2×Q16, C5⋊2C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C8.2D4, C40⋊C2 [×2], C2×C5⋊2C8, C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×Q16 [×2], C2×Dic10, C2×D20, Q8×C10 [×2], C40⋊8C4, C2×C40⋊C2, C2×Q8⋊D5, C2×C5⋊Q16, Dic5⋊Q8, C20.23D4, C10×Q16, C40.37D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C4⋊1D4, C8.C22 [×2], C5⋊D4 [×2], C22×D5, C8.2D4, D4×D5 [×2], C2×C5⋊D4, Q16⋊D5 [×2], C20⋊D4, C40.37D4
►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)
(1 113 137 53)(2 102 138 42)(3 91 139 71)(4 120 140 60)(5 109 141 49)(6 98 142 78)(7 87 143 67)(8 116 144 56)(9 105 145 45)(10 94 146 74)(11 83 147 63)(12 112 148 52)(13 101 149 41)(14 90 150 70)(15 119 151 59)(16 108 152 48)(17 97 153 77)(18 86 154 66)(19 115 155 55)(20 104 156 44)(21 93 157 73)(22 82 158 62)(23 111 159 51)(24 100 160 80)(25 89 121 69)(26 118 122 58)(27 107 123 47)(28 96 124 76)(29 85 125 65)(30 114 126 54)(31 103 127 43)(32 92 128 72)(33 81 129 61)(34 110 130 50)(35 99 131 79)(36 88 132 68)(37 117 133 57)(38 106 134 46)(39 95 135 75)(40 84 136 64)
(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)(41 105)(42 84)(43 103)(44 82)(45 101)(46 120)(47 99)(48 118)(49 97)(50 116)(51 95)(52 114)(53 93)(54 112)(55 91)(56 110)(57 89)(58 108)(59 87)(60 106)(61 85)(62 104)(63 83)(64 102)(65 81)(66 100)(67 119)(68 98)(69 117)(70 96)(71 115)(72 94)(73 113)(74 92)(75 111)(76 90)(77 109)(78 88)(79 107)(80 86)(121 153)(122 132)(123 151)(124 130)(125 149)(126 128)(127 147)(129 145)(131 143)(133 141)(134 160)(135 139)(136 158)(138 156)(140 154)(142 152)(144 150)(146 148)(155 159)
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), (1,113,137,53)(2,102,138,42)(3,91,139,71)(4,120,140,60)(5,109,141,49)(6,98,142,78)(7,87,143,67)(8,116,144,56)(9,105,145,45)(10,94,146,74)(11,83,147,63)(12,112,148,52)(13,101,149,41)(14,90,150,70)(15,119,151,59)(16,108,152,48)(17,97,153,77)(18,86,154,66)(19,115,155,55)(20,104,156,44)(21,93,157,73)(22,82,158,62)(23,111,159,51)(24,100,160,80)(25,89,121,69)(26,118,122,58)(27,107,123,47)(28,96,124,76)(29,85,125,65)(30,114,126,54)(31,103,127,43)(32,92,128,72)(33,81,129,61)(34,110,130,50)(35,99,131,79)(36,88,132,68)(37,117,133,57)(38,106,134,46)(39,95,135,75)(40,84,136,64), (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)(41,105)(42,84)(43,103)(44,82)(45,101)(46,120)(47,99)(48,118)(49,97)(50,116)(51,95)(52,114)(53,93)(54,112)(55,91)(56,110)(57,89)(58,108)(59,87)(60,106)(61,85)(62,104)(63,83)(64,102)(65,81)(66,100)(67,119)(68,98)(69,117)(70,96)(71,115)(72,94)(73,113)(74,92)(75,111)(76,90)(77,109)(78,88)(79,107)(80,86)(121,153)(122,132)(123,151)(124,130)(125,149)(126,128)(127,147)(129,145)(131,143)(133,141)(134,160)(135,139)(136,158)(138,156)(140,154)(142,152)(144,150)(146,148)(155,159)>;
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), (1,113,137,53)(2,102,138,42)(3,91,139,71)(4,120,140,60)(5,109,141,49)(6,98,142,78)(7,87,143,67)(8,116,144,56)(9,105,145,45)(10,94,146,74)(11,83,147,63)(12,112,148,52)(13,101,149,41)(14,90,150,70)(15,119,151,59)(16,108,152,48)(17,97,153,77)(18,86,154,66)(19,115,155,55)(20,104,156,44)(21,93,157,73)(22,82,158,62)(23,111,159,51)(24,100,160,80)(25,89,121,69)(26,118,122,58)(27,107,123,47)(28,96,124,76)(29,85,125,65)(30,114,126,54)(31,103,127,43)(32,92,128,72)(33,81,129,61)(34,110,130,50)(35,99,131,79)(36,88,132,68)(37,117,133,57)(38,106,134,46)(39,95,135,75)(40,84,136,64), (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)(41,105)(42,84)(43,103)(44,82)(45,101)(46,120)(47,99)(48,118)(49,97)(50,116)(51,95)(52,114)(53,93)(54,112)(55,91)(56,110)(57,89)(58,108)(59,87)(60,106)(61,85)(62,104)(63,83)(64,102)(65,81)(66,100)(67,119)(68,98)(69,117)(70,96)(71,115)(72,94)(73,113)(74,92)(75,111)(76,90)(77,109)(78,88)(79,107)(80,86)(121,153)(122,132)(123,151)(124,130)(125,149)(126,128)(127,147)(129,145)(131,143)(133,141)(134,160)(135,139)(136,158)(138,156)(140,154)(142,152)(144,150)(146,148)(155,159) );
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)], [(1,113,137,53),(2,102,138,42),(3,91,139,71),(4,120,140,60),(5,109,141,49),(6,98,142,78),(7,87,143,67),(8,116,144,56),(9,105,145,45),(10,94,146,74),(11,83,147,63),(12,112,148,52),(13,101,149,41),(14,90,150,70),(15,119,151,59),(16,108,152,48),(17,97,153,77),(18,86,154,66),(19,115,155,55),(20,104,156,44),(21,93,157,73),(22,82,158,62),(23,111,159,51),(24,100,160,80),(25,89,121,69),(26,118,122,58),(27,107,123,47),(28,96,124,76),(29,85,125,65),(30,114,126,54),(31,103,127,43),(32,92,128,72),(33,81,129,61),(34,110,130,50),(35,99,131,79),(36,88,132,68),(37,117,133,57),(38,106,134,46),(39,95,135,75),(40,84,136,64)], [(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),(41,105),(42,84),(43,103),(44,82),(45,101),(46,120),(47,99),(48,118),(49,97),(50,116),(51,95),(52,114),(53,93),(54,112),(55,91),(56,110),(57,89),(58,108),(59,87),(60,106),(61,85),(62,104),(63,83),(64,102),(65,81),(66,100),(67,119),(68,98),(69,117),(70,96),(71,115),(72,94),(73,113),(74,92),(75,111),(76,90),(77,109),(78,88),(79,107),(80,86),(121,153),(122,132),(123,151),(124,130),(125,149),(126,128),(127,147),(129,145),(131,143),(133,141),(134,160),(135,139),(136,158),(138,156),(140,154),(142,152),(144,150),(146,148),(155,159)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 8 | 8 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C5⋊D4 | C8.C22 | D4×D5 | D4×D5 | Q16⋊D5 |
| kernel | C40.37D4 | C40⋊8C4 | C2×C40⋊C2 | C2×Q8⋊D5 | C2×C5⋊Q16 | Dic5⋊Q8 | C20.23D4 | C10×Q16 | C5⋊2C8 | C40 | C2×Dic5 | C2×Q16 | C2×C8 | C2×Q8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.37D4 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 36 | 7 | 7 |
| 0 | 0 | 0 | 0 | 10 | 18 | 0 | 24 |
| 0 | 0 | 0 | 0 | 19 | 21 | 3 | 3 |
| 0 | 0 | 0 | 0 | 29 | 21 | 21 | 3 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 40 | 40 |
| 0 | 0 | 0 | 0 | 21 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,17,10,19,29,0,0,0,0,36,18,21,21,0,0,0,0,7,0,3,21,0,0,0,0,7,24,3,3],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,1,21,21,0,0,0,0,0,0,0,0,1,0,0,0,0,37,40,40,40,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,1,21,0,21,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;
C40.37D4 in GAP, Magma, Sage, TeX
C_{40}._{37}D_4 % in TeX
G:=Group("C40.37D4"); // GroupNames label
G:=SmallGroup(320,817);
// by ID
G=gap.SmallGroup(320,817);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,219,184,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=a^20*b^-1>;
// generators/relations