metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊15D4, Dic5⋊1SD16, C5⋊2C8⋊15D4, C8⋊8(C5⋊D4), C5⋊4(C8⋊5D4), C4.25(D4×D5), C20.50(C2×D4), (C10×SD16)⋊9C2, (C8×Dic5)⋊11C2, (C2×SD16)⋊15D5, (C2×D4).76D10, (C2×C8).264D10, C20⋊D4.8C2, (C2×Q8).57D10, C2.31(D5×SD16), Dic5⋊Q8⋊5C2, C10.48(C2×SD16), C22.272(D4×D5), C2.22(C20⋊D4), C10.31(C4⋊1D4), (C2×C20).452C23, (C2×C40).165C22, (C2×Dic5).160D4, (Q8×C10).81C22, (D4×C10).101C22, (C2×D20).126C22, (C4×Dic5).273C22, (C2×Dic10).133C22, C4.9(C2×C5⋊D4), (C2×Q8⋊D5)⋊18C2, (C2×C40⋊C2)⋊30C2, (C2×D4.D5)⋊21C2, (C2×C10).364(C2×D4), (C2×C4).541(C22×D5), (C2×C5⋊2C8).283C22, SmallGroup(320,802)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊15D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a19, cbc=b-1 >
Subgroups: 654 in 142 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], Q8 [×4], C23 [×2], D5, C10, C10 [×2], C10, C42, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×8], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, Dic5 [×4], Dic5, C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C2×SD16 [×3], C5⋊2C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C8⋊5D4, C40⋊C2 [×2], C2×C5⋊2C8, C4×Dic5, C10.D4 [×2], D4.D5 [×2], Q8⋊D5 [×2], C2×C40, C5×SD16 [×2], C2×Dic10, C2×D20, C2×C5⋊D4 [×2], D4×C10, Q8×C10, C8×Dic5, C2×C40⋊C2, C2×D4.D5, C20⋊D4, C2×Q8⋊D5, Dic5⋊Q8, C10×SD16, C40⋊15D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×4], C2×D4 [×3], D10 [×3], C4⋊1D4, C2×SD16 [×2], C5⋊D4 [×2], C22×D5, C8⋊5D4, D4×D5 [×2], C2×C5⋊D4, D5×SD16 [×2], C20⋊D4, C40⋊15D4
►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)
(1 104 149 54)(2 113 150 63)(3 82 151 72)(4 91 152 41)(5 100 153 50)(6 109 154 59)(7 118 155 68)(8 87 156 77)(9 96 157 46)(10 105 158 55)(11 114 159 64)(12 83 160 73)(13 92 121 42)(14 101 122 51)(15 110 123 60)(16 119 124 69)(17 88 125 78)(18 97 126 47)(19 106 127 56)(20 115 128 65)(21 84 129 74)(22 93 130 43)(23 102 131 52)(24 111 132 61)(25 120 133 70)(26 89 134 79)(27 98 135 48)(28 107 136 57)(29 116 137 66)(30 85 138 75)(31 94 139 44)(32 103 140 53)(33 112 141 62)(34 81 142 71)(35 90 143 80)(36 99 144 49)(37 108 145 58)(38 117 146 67)(39 86 147 76)(40 95 148 45)
(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 97)(42 116)(43 95)(44 114)(45 93)(46 112)(47 91)(48 110)(49 89)(50 108)(51 87)(52 106)(53 85)(54 104)(55 83)(56 102)(57 81)(58 100)(59 119)(60 98)(61 117)(62 96)(63 115)(64 94)(65 113)(66 92)(67 111)(68 90)(69 109)(70 88)(71 107)(72 86)(73 105)(74 84)(75 103)(76 82)(77 101)(78 120)(79 99)(80 118)(121 137)(122 156)(123 135)(124 154)(125 133)(126 152)(127 131)(128 150)(130 148)(132 146)(134 144)(136 142)(138 140)(139 159)(141 157)(143 155)(145 153)(147 151)(158 160)
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), (1,104,149,54)(2,113,150,63)(3,82,151,72)(4,91,152,41)(5,100,153,50)(6,109,154,59)(7,118,155,68)(8,87,156,77)(9,96,157,46)(10,105,158,55)(11,114,159,64)(12,83,160,73)(13,92,121,42)(14,101,122,51)(15,110,123,60)(16,119,124,69)(17,88,125,78)(18,97,126,47)(19,106,127,56)(20,115,128,65)(21,84,129,74)(22,93,130,43)(23,102,131,52)(24,111,132,61)(25,120,133,70)(26,89,134,79)(27,98,135,48)(28,107,136,57)(29,116,137,66)(30,85,138,75)(31,94,139,44)(32,103,140,53)(33,112,141,62)(34,81,142,71)(35,90,143,80)(36,99,144,49)(37,108,145,58)(38,117,146,67)(39,86,147,76)(40,95,148,45), (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)(41,97)(42,116)(43,95)(44,114)(45,93)(46,112)(47,91)(48,110)(49,89)(50,108)(51,87)(52,106)(53,85)(54,104)(55,83)(56,102)(57,81)(58,100)(59,119)(60,98)(61,117)(62,96)(63,115)(64,94)(65,113)(66,92)(67,111)(68,90)(69,109)(70,88)(71,107)(72,86)(73,105)(74,84)(75,103)(76,82)(77,101)(78,120)(79,99)(80,118)(121,137)(122,156)(123,135)(124,154)(125,133)(126,152)(127,131)(128,150)(130,148)(132,146)(134,144)(136,142)(138,140)(139,159)(141,157)(143,155)(145,153)(147,151)(158,160)>;
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), (1,104,149,54)(2,113,150,63)(3,82,151,72)(4,91,152,41)(5,100,153,50)(6,109,154,59)(7,118,155,68)(8,87,156,77)(9,96,157,46)(10,105,158,55)(11,114,159,64)(12,83,160,73)(13,92,121,42)(14,101,122,51)(15,110,123,60)(16,119,124,69)(17,88,125,78)(18,97,126,47)(19,106,127,56)(20,115,128,65)(21,84,129,74)(22,93,130,43)(23,102,131,52)(24,111,132,61)(25,120,133,70)(26,89,134,79)(27,98,135,48)(28,107,136,57)(29,116,137,66)(30,85,138,75)(31,94,139,44)(32,103,140,53)(33,112,141,62)(34,81,142,71)(35,90,143,80)(36,99,144,49)(37,108,145,58)(38,117,146,67)(39,86,147,76)(40,95,148,45), (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)(41,97)(42,116)(43,95)(44,114)(45,93)(46,112)(47,91)(48,110)(49,89)(50,108)(51,87)(52,106)(53,85)(54,104)(55,83)(56,102)(57,81)(58,100)(59,119)(60,98)(61,117)(62,96)(63,115)(64,94)(65,113)(66,92)(67,111)(68,90)(69,109)(70,88)(71,107)(72,86)(73,105)(74,84)(75,103)(76,82)(77,101)(78,120)(79,99)(80,118)(121,137)(122,156)(123,135)(124,154)(125,133)(126,152)(127,131)(128,150)(130,148)(132,146)(134,144)(136,142)(138,140)(139,159)(141,157)(143,155)(145,153)(147,151)(158,160) );
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)], [(1,104,149,54),(2,113,150,63),(3,82,151,72),(4,91,152,41),(5,100,153,50),(6,109,154,59),(7,118,155,68),(8,87,156,77),(9,96,157,46),(10,105,158,55),(11,114,159,64),(12,83,160,73),(13,92,121,42),(14,101,122,51),(15,110,123,60),(16,119,124,69),(17,88,125,78),(18,97,126,47),(19,106,127,56),(20,115,128,65),(21,84,129,74),(22,93,130,43),(23,102,131,52),(24,111,132,61),(25,120,133,70),(26,89,134,79),(27,98,135,48),(28,107,136,57),(29,116,137,66),(30,85,138,75),(31,94,139,44),(32,103,140,53),(33,112,141,62),(34,81,142,71),(35,90,143,80),(36,99,144,49),(37,108,145,58),(38,117,146,67),(39,86,147,76),(40,95,148,45)], [(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),(41,97),(42,116),(43,95),(44,114),(45,93),(46,112),(47,91),(48,110),(49,89),(50,108),(51,87),(52,106),(53,85),(54,104),(55,83),(56,102),(57,81),(58,100),(59,119),(60,98),(61,117),(62,96),(63,115),(64,94),(65,113),(66,92),(67,111),(68,90),(69,109),(70,88),(71,107),(72,86),(73,105),(74,84),(75,103),(76,82),(77,101),(78,120),(79,99),(80,118),(121,137),(122,156),(123,135),(124,154),(125,133),(126,152),(127,131),(128,150),(130,148),(132,146),(134,144),(136,142),(138,140),(139,159),(141,157),(143,155),(145,153),(147,151),(158,160)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 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 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C5⋊D4 | D4×D5 | D4×D5 | D5×SD16 |
| kernel | C40⋊15D4 | C8×Dic5 | C2×C40⋊C2 | C2×D4.D5 | C20⋊D4 | C2×Q8⋊D5 | Dic5⋊Q8 | C10×SD16 | C5⋊2C8 | C40 | C2×Dic5 | C2×SD16 | Dic5 | C2×C8 | C2×D4 | C2×Q8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 8 | 2 | 2 | 8 |
Matrix representation of C40⋊15D4 ►in GL6(𝔽41)
| 0 | 35 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [0,7,0,0,0,0,35,7,0,0,0,0,0,0,15,15,0,0,0,0,26,15,0,0,0,0,0,0,15,15,0,0,0,0,26,15],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C40⋊15D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{15}D_4 % in TeX
G:=Group("C40:15D4"); // GroupNames label
G:=SmallGroup(320,802);
// by ID
G=gap.SmallGroup(320,802);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations