metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.29D4, D20.27D4, Dic10.27D4, C4.69(D4×D5), (C2×Q16)⋊10D5, (C10×Q16)⋊2C2, (C2×C8).97D10, C40.6C4⋊4C2, C20.190(C2×D4), C5⋊4(D4.5D4), C8.29(C5⋊D4), (C2×Q8).68D10, C20.10D4⋊8C2, (C2×C40).34C22, C2.24(C20⋊2D4), (C2×C20).466C23, D20.3C4.1C2, C20.C23.2C2, C4○D20.48C22, (Q8×C10).95C22, C10.123(C4⋊D4), C4.Dic5.21C22, C22.22(D4⋊2D5), C4.87(C2×C5⋊D4), (C2×C4).128(C22×D5), (C2×C10).160(C4○D4), SmallGroup(320,819)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.29D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=a-1, cac=a9, cbc=b3 >
Subgroups: 334 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], D5, C10, C10, C2×C8, C2×C8, M4(2) [×4], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C5⋊2C8 [×3], C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20 [×2], C5×Q8 [×4], D4.5D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×Q16 [×2], C4○D20, Q8×C10 [×2], C40.6C4, C20.10D4 [×2], D20.3C4, C20.C23 [×2], C10×Q16, C40.29D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C5⋊D4 [×2], C22×D5, D4.5D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4, C40.29D4
►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 61 116 159 21 41 96 139)(2 60 117 158 22 80 97 138)(3 59 118 157 23 79 98 137)(4 58 119 156 24 78 99 136)(5 57 120 155 25 77 100 135)(6 56 81 154 26 76 101 134)(7 55 82 153 27 75 102 133)(8 54 83 152 28 74 103 132)(9 53 84 151 29 73 104 131)(10 52 85 150 30 72 105 130)(11 51 86 149 31 71 106 129)(12 50 87 148 32 70 107 128)(13 49 88 147 33 69 108 127)(14 48 89 146 34 68 109 126)(15 47 90 145 35 67 110 125)(16 46 91 144 36 66 111 124)(17 45 92 143 37 65 112 123)(18 44 93 142 38 64 113 122)(19 43 94 141 39 63 114 121)(20 42 95 140 40 62 115 160)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 139)(42 148)(43 157)(44 126)(45 135)(46 144)(47 153)(48 122)(49 131)(50 140)(51 149)(52 158)(53 127)(54 136)(55 145)(56 154)(57 123)(58 132)(59 141)(60 150)(61 159)(62 128)(63 137)(64 146)(65 155)(66 124)(67 133)(68 142)(69 151)(70 160)(71 129)(72 138)(73 147)(74 156)(75 125)(76 134)(77 143)(78 152)(79 121)(80 130)(81 101)(82 110)(83 119)(84 88)(85 97)(86 106)(87 115)(89 93)(90 102)(91 111)(92 120)(94 98)(95 107)(96 116)(99 103)(100 112)(104 108)(105 117)(109 113)(114 118)
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), (1,61,116,159,21,41,96,139)(2,60,117,158,22,80,97,138)(3,59,118,157,23,79,98,137)(4,58,119,156,24,78,99,136)(5,57,120,155,25,77,100,135)(6,56,81,154,26,76,101,134)(7,55,82,153,27,75,102,133)(8,54,83,152,28,74,103,132)(9,53,84,151,29,73,104,131)(10,52,85,150,30,72,105,130)(11,51,86,149,31,71,106,129)(12,50,87,148,32,70,107,128)(13,49,88,147,33,69,108,127)(14,48,89,146,34,68,109,126)(15,47,90,145,35,67,110,125)(16,46,91,144,36,66,111,124)(17,45,92,143,37,65,112,123)(18,44,93,142,38,64,113,122)(19,43,94,141,39,63,114,121)(20,42,95,140,40,62,115,160), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(41,139)(42,148)(43,157)(44,126)(45,135)(46,144)(47,153)(48,122)(49,131)(50,140)(51,149)(52,158)(53,127)(54,136)(55,145)(56,154)(57,123)(58,132)(59,141)(60,150)(61,159)(62,128)(63,137)(64,146)(65,155)(66,124)(67,133)(68,142)(69,151)(70,160)(71,129)(72,138)(73,147)(74,156)(75,125)(76,134)(77,143)(78,152)(79,121)(80,130)(81,101)(82,110)(83,119)(84,88)(85,97)(86,106)(87,115)(89,93)(90,102)(91,111)(92,120)(94,98)(95,107)(96,116)(99,103)(100,112)(104,108)(105,117)(109,113)(114,118)>;
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), (1,61,116,159,21,41,96,139)(2,60,117,158,22,80,97,138)(3,59,118,157,23,79,98,137)(4,58,119,156,24,78,99,136)(5,57,120,155,25,77,100,135)(6,56,81,154,26,76,101,134)(7,55,82,153,27,75,102,133)(8,54,83,152,28,74,103,132)(9,53,84,151,29,73,104,131)(10,52,85,150,30,72,105,130)(11,51,86,149,31,71,106,129)(12,50,87,148,32,70,107,128)(13,49,88,147,33,69,108,127)(14,48,89,146,34,68,109,126)(15,47,90,145,35,67,110,125)(16,46,91,144,36,66,111,124)(17,45,92,143,37,65,112,123)(18,44,93,142,38,64,113,122)(19,43,94,141,39,63,114,121)(20,42,95,140,40,62,115,160), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(41,139)(42,148)(43,157)(44,126)(45,135)(46,144)(47,153)(48,122)(49,131)(50,140)(51,149)(52,158)(53,127)(54,136)(55,145)(56,154)(57,123)(58,132)(59,141)(60,150)(61,159)(62,128)(63,137)(64,146)(65,155)(66,124)(67,133)(68,142)(69,151)(70,160)(71,129)(72,138)(73,147)(74,156)(75,125)(76,134)(77,143)(78,152)(79,121)(80,130)(81,101)(82,110)(83,119)(84,88)(85,97)(86,106)(87,115)(89,93)(90,102)(91,111)(92,120)(94,98)(95,107)(96,116)(99,103)(100,112)(104,108)(105,117)(109,113)(114,118) );
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)], [(1,61,116,159,21,41,96,139),(2,60,117,158,22,80,97,138),(3,59,118,157,23,79,98,137),(4,58,119,156,24,78,99,136),(5,57,120,155,25,77,100,135),(6,56,81,154,26,76,101,134),(7,55,82,153,27,75,102,133),(8,54,83,152,28,74,103,132),(9,53,84,151,29,73,104,131),(10,52,85,150,30,72,105,130),(11,51,86,149,31,71,106,129),(12,50,87,148,32,70,107,128),(13,49,88,147,33,69,108,127),(14,48,89,146,34,68,109,126),(15,47,90,145,35,67,110,125),(16,46,91,144,36,66,111,124),(17,45,92,143,37,65,112,123),(18,44,93,142,38,64,113,122),(19,43,94,141,39,63,114,121),(20,42,95,140,40,62,115,160)], [(2,10),(3,19),(4,28),(5,37),(7,15),(8,24),(9,33),(12,20),(13,29),(14,38),(17,25),(18,34),(22,30),(23,39),(27,35),(32,40),(41,139),(42,148),(43,157),(44,126),(45,135),(46,144),(47,153),(48,122),(49,131),(50,140),(51,149),(52,158),(53,127),(54,136),(55,145),(56,154),(57,123),(58,132),(59,141),(60,150),(61,159),(62,128),(63,137),(64,146),(65,155),(66,124),(67,133),(68,142),(69,151),(70,160),(71,129),(72,138),(73,147),(74,156),(75,125),(76,134),(77,143),(78,152),(79,121),(80,130),(81,101),(82,110),(83,119),(84,88),(85,97),(86,106),(87,115),(89,93),(90,102),(91,111),(92,120),(94,98),(95,107),(96,116),(99,103),(100,112),(104,108),(105,117),(109,113),(114,118)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 2 | 2 | 8 | 8 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 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 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D4.5D4 | D4×D5 | D4⋊2D5 | C40.29D4 |
| kernel | C40.29D4 | C40.6C4 | C20.10D4 | D20.3C4 | C20.C23 | C10×Q16 | C40 | Dic10 | D20 | C2×Q16 | C2×C10 | C2×C8 | C2×Q8 | C8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.29D4 ►in GL4(𝔽41) generated by
| 22 | 20 | 31 | 21 |
| 0 | 0 | 31 | 40 |
| 31 | 2 | 4 | 27 |
| 19 | 21 | 21 | 19 |
| 31 | 7 | 6 | 32 |
| 9 | 20 | 11 | 12 |
| 32 | 24 | 15 | 39 |
| 31 | 40 | 21 | 16 |
| 6 | 40 | 6 | 40 |
| 35 | 35 | 0 | 35 |
| 0 | 0 | 34 | 8 |
| 0 | 0 | 35 | 7 |
G:=sub<GL(4,GF(41))| [22,0,31,19,20,0,2,21,31,31,4,21,21,40,27,19],[31,9,32,31,7,20,24,40,6,11,15,21,32,12,39,16],[6,35,0,0,40,35,0,0,6,0,34,35,40,35,8,7] >;
C40.29D4 in GAP, Magma, Sage, TeX
C_{40}._{29}D_4 % in TeX
G:=Group("C40.29D4"); // GroupNames label
G:=SmallGroup(320,819);
// by ID
G=gap.SmallGroup(320,819);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,1123,297,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^3>;
// generators/relations