metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.28D4, (C2×Q16)⋊6D5, C4.29(D4×D5), (C10×Q16)⋊6C2, (C8×Dic5)⋊7C2, C5⋊2C8.34D4, (C2×D40).11C2, (C2×C8).244D10, C20.189(C2×D4), C5⋊5(C8.12D4), C8.19(C5⋊D4), (C2×Q8).67D10, C10.82(C4○D8), C20.23D4⋊6C2, (C2×C40).96C22, C22.282(D4×D5), C2.26(C20⋊D4), C10.35(C4⋊1D4), (C2×C20).465C23, (C2×Dic5).163D4, (Q8×C10).94C22, C2.19(Q8.D10), (C2×D20).131C22, (C4×Dic5).276C22, (C2×Q8⋊D5)⋊21C2, C4.16(C2×C5⋊D4), (C2×C10).376(C2×D4), (C2×C4).553(C22×D5), (C2×C5⋊2C8).287C22, SmallGroup(320,818)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.28D4
G = < a,b,c | a40=b4=c2=1, bab-1=a9, cac=a-1, cbc=a20b-1 >
Subgroups: 622 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C5⋊2C8, C40, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C8.12D4, D40, C2×C5⋊2C8, C4×Dic5, D10⋊C4, Q8⋊D5, C2×C40, C5×Q16, C2×D20, Q8×C10, C8×Dic5, C2×D40, C2×Q8⋊D5, C20.23D4, C10×Q16, C40.28D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C4○D8, C5⋊D4, C22×D5, C8.12D4, D4×D5, C2×C5⋊D4, Q8.D10, C20⋊D4, C40.28D4
►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)
(1 121 53 88)(2 130 54 97)(3 139 55 106)(4 148 56 115)(5 157 57 84)(6 126 58 93)(7 135 59 102)(8 144 60 111)(9 153 61 120)(10 122 62 89)(11 131 63 98)(12 140 64 107)(13 149 65 116)(14 158 66 85)(15 127 67 94)(16 136 68 103)(17 145 69 112)(18 154 70 81)(19 123 71 90)(20 132 72 99)(21 141 73 108)(22 150 74 117)(23 159 75 86)(24 128 76 95)(25 137 77 104)(26 146 78 113)(27 155 79 82)(28 124 80 91)(29 133 41 100)(30 142 42 109)(31 151 43 118)(32 160 44 87)(33 129 45 96)(34 138 46 105)(35 147 47 114)(36 156 48 83)(37 125 49 92)(38 134 50 101)(39 143 51 110)(40 152 52 119)
(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 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 148)(82 147)(83 146)(84 145)(85 144)(86 143)(87 142)(88 141)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 160)(110 159)(111 158)(112 157)(113 156)(114 155)(115 154)(116 153)(117 152)(118 151)(119 150)(120 149)
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), (1,121,53,88)(2,130,54,97)(3,139,55,106)(4,148,56,115)(5,157,57,84)(6,126,58,93)(7,135,59,102)(8,144,60,111)(9,153,61,120)(10,122,62,89)(11,131,63,98)(12,140,64,107)(13,149,65,116)(14,158,66,85)(15,127,67,94)(16,136,68,103)(17,145,69,112)(18,154,70,81)(19,123,71,90)(20,132,72,99)(21,141,73,108)(22,150,74,117)(23,159,75,86)(24,128,76,95)(25,137,77,104)(26,146,78,113)(27,155,79,82)(28,124,80,91)(29,133,41,100)(30,142,42,109)(31,151,43,118)(32,160,44,87)(33,129,45,96)(34,138,46,105)(35,147,47,114)(36,156,48,83)(37,125,49,92)(38,134,50,101)(39,143,51,110)(40,152,52,119), (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)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,148)(82,147)(83,146)(84,145)(85,144)(86,143)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,160)(110,159)(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149)>;
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), (1,121,53,88)(2,130,54,97)(3,139,55,106)(4,148,56,115)(5,157,57,84)(6,126,58,93)(7,135,59,102)(8,144,60,111)(9,153,61,120)(10,122,62,89)(11,131,63,98)(12,140,64,107)(13,149,65,116)(14,158,66,85)(15,127,67,94)(16,136,68,103)(17,145,69,112)(18,154,70,81)(19,123,71,90)(20,132,72,99)(21,141,73,108)(22,150,74,117)(23,159,75,86)(24,128,76,95)(25,137,77,104)(26,146,78,113)(27,155,79,82)(28,124,80,91)(29,133,41,100)(30,142,42,109)(31,151,43,118)(32,160,44,87)(33,129,45,96)(34,138,46,105)(35,147,47,114)(36,156,48,83)(37,125,49,92)(38,134,50,101)(39,143,51,110)(40,152,52,119), (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)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(81,148)(82,147)(83,146)(84,145)(85,144)(86,143)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,160)(110,159)(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)(119,150)(120,149) );
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)], [(1,121,53,88),(2,130,54,97),(3,139,55,106),(4,148,56,115),(5,157,57,84),(6,126,58,93),(7,135,59,102),(8,144,60,111),(9,153,61,120),(10,122,62,89),(11,131,63,98),(12,140,64,107),(13,149,65,116),(14,158,66,85),(15,127,67,94),(16,136,68,103),(17,145,69,112),(18,154,70,81),(19,123,71,90),(20,132,72,99),(21,141,73,108),(22,150,74,117),(23,159,75,86),(24,128,76,95),(25,137,77,104),(26,146,78,113),(27,155,79,82),(28,124,80,91),(29,133,41,100),(30,142,42,109),(31,151,43,118),(32,160,44,87),(33,129,45,96),(34,138,46,105),(35,147,47,114),(36,156,48,83),(37,125,49,92),(38,134,50,101),(39,143,51,110),(40,152,52,119)], [(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),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(81,148),(82,147),(83,146),(84,145),(85,144),(86,143),(87,142),(88,141),(89,140),(90,139),(91,138),(92,137),(93,136),(94,135),(95,134),(96,133),(97,132),(98,131),(99,130),(100,129),(101,128),(102,127),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,160),(110,159),(111,158),(112,157),(113,156),(114,155),(115,154),(116,153),(117,152),(118,151),(119,150),(120,149)]])
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 | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 8 | 8 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C5⋊D4 | D4×D5 | D4×D5 | Q8.D10 |
| kernel | C40.28D4 | C8×Dic5 | C2×D40 | C2×Q8⋊D5 | C20.23D4 | C10×Q16 | C5⋊2C8 | C40 | C2×Dic5 | C2×Q16 | C2×C8 | C2×Q8 | C10 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 8 |
Matrix representation of C40.28D4 ►in GL4(𝔽41) generated by
| 34 | 7 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 15 | 17 |
| 3 | 3 | 0 | 0 |
| 24 | 38 | 0 | 0 |
| 0 | 0 | 32 | 39 |
| 0 | 0 | 40 | 9 |
| 34 | 7 | 0 | 0 |
| 40 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 9 | 1 |
G:=sub<GL(4,GF(41))| [34,34,0,0,7,1,0,0,0,0,0,15,0,0,30,17],[3,24,0,0,3,38,0,0,0,0,32,40,0,0,39,9],[34,40,0,0,7,7,0,0,0,0,40,9,0,0,0,1] >;
C40.28D4 in GAP, Magma, Sage, TeX
C_{40}._{28}D_4 % in TeX
G:=Group("C40.28D4"); // GroupNames label
G:=SmallGroup(320,818);
// by ID
G=gap.SmallGroup(320,818);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,184,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^20*b^-1>;
// generators/relations