metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.36D4, (C2×Q16)⋊8D5, (C10×Q16)⋊9C2, C40⋊6C4⋊22C2, (C2×C8).95D10, C5⋊5(C8.D4), C8.5(C5⋊D4), C20.187(C2×D4), (C2×Q8).65D10, Q8⋊Dic5⋊36C2, (C2×Dic5).85D4, (C22×D5).49D4, C22.280(D4×D5), C20.108(C4○D4), C4.37(D4⋊2D5), C2.23(C20⋊2D4), (C2×C20).463C23, (C2×C40).150C22, D10⋊3Q8.10C2, (Q8×C10).92C22, C10.122(C4⋊D4), C2.30(Q16⋊D5), C10.80(C8.C22), C4⋊Dic5.186C22, C4.86(C2×C5⋊D4), (C2×C8⋊D5).5C2, (C2×C4×D5).58C22, (C2×C10).374(C2×D4), (C2×C4).551(C22×D5), (C2×C5⋊2C8).167C22, SmallGroup(320,816)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.36D4
G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a29, cbc=a20b-1 >
Subgroups: 406 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C8.D4, C8⋊D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C2×C40, C5×Q16, C2×C4×D5, Q8×C10, C40⋊6C4, Q8⋊Dic5, C2×C8⋊D5, D10⋊3Q8, C10×Q16, C40.36D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, C5⋊D4, C22×D5, C8.D4, D4×D5, D4⋊2D5, C2×C5⋊D4, Q16⋊D5, C20⋊2D4, C40.36D4
►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)
(1 126 47 88)(2 145 48 107)(3 124 49 86)(4 143 50 105)(5 122 51 84)(6 141 52 103)(7 160 53 82)(8 139 54 101)(9 158 55 120)(10 137 56 99)(11 156 57 118)(12 135 58 97)(13 154 59 116)(14 133 60 95)(15 152 61 114)(16 131 62 93)(17 150 63 112)(18 129 64 91)(19 148 65 110)(20 127 66 89)(21 146 67 108)(22 125 68 87)(23 144 69 106)(24 123 70 85)(25 142 71 104)(26 121 72 83)(27 140 73 102)(28 159 74 81)(29 138 75 100)(30 157 76 119)(31 136 77 98)(32 155 78 117)(33 134 79 96)(34 153 80 115)(35 132 41 94)(36 151 42 113)(37 130 43 92)(38 149 44 111)(39 128 45 90)(40 147 46 109)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 73)(42 62)(43 51)(44 80)(45 69)(46 58)(48 76)(49 65)(50 54)(52 72)(53 61)(55 79)(56 68)(59 75)(60 64)(63 71)(66 78)(70 74)(81 143)(82 132)(83 121)(84 150)(85 139)(86 128)(87 157)(88 146)(89 135)(90 124)(91 153)(92 142)(93 131)(94 160)(95 149)(96 138)(97 127)(98 156)(99 145)(100 134)(101 123)(102 152)(103 141)(104 130)(105 159)(106 148)(107 137)(108 126)(109 155)(110 144)(111 133)(112 122)(113 151)(114 140)(115 129)(116 158)(117 147)(118 136)(119 125)(120 154)
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), (1,126,47,88)(2,145,48,107)(3,124,49,86)(4,143,50,105)(5,122,51,84)(6,141,52,103)(7,160,53,82)(8,139,54,101)(9,158,55,120)(10,137,56,99)(11,156,57,118)(12,135,58,97)(13,154,59,116)(14,133,60,95)(15,152,61,114)(16,131,62,93)(17,150,63,112)(18,129,64,91)(19,148,65,110)(20,127,66,89)(21,146,67,108)(22,125,68,87)(23,144,69,106)(24,123,70,85)(25,142,71,104)(26,121,72,83)(27,140,73,102)(28,159,74,81)(29,138,75,100)(30,157,76,119)(31,136,77,98)(32,155,78,117)(33,134,79,96)(34,153,80,115)(35,132,41,94)(36,151,42,113)(37,130,43,92)(38,149,44,111)(39,128,45,90)(40,147,46,109), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,73)(42,62)(43,51)(44,80)(45,69)(46,58)(48,76)(49,65)(50,54)(52,72)(53,61)(55,79)(56,68)(59,75)(60,64)(63,71)(66,78)(70,74)(81,143)(82,132)(83,121)(84,150)(85,139)(86,128)(87,157)(88,146)(89,135)(90,124)(91,153)(92,142)(93,131)(94,160)(95,149)(96,138)(97,127)(98,156)(99,145)(100,134)(101,123)(102,152)(103,141)(104,130)(105,159)(106,148)(107,137)(108,126)(109,155)(110,144)(111,133)(112,122)(113,151)(114,140)(115,129)(116,158)(117,147)(118,136)(119,125)(120,154)>;
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), (1,126,47,88)(2,145,48,107)(3,124,49,86)(4,143,50,105)(5,122,51,84)(6,141,52,103)(7,160,53,82)(8,139,54,101)(9,158,55,120)(10,137,56,99)(11,156,57,118)(12,135,58,97)(13,154,59,116)(14,133,60,95)(15,152,61,114)(16,131,62,93)(17,150,63,112)(18,129,64,91)(19,148,65,110)(20,127,66,89)(21,146,67,108)(22,125,68,87)(23,144,69,106)(24,123,70,85)(25,142,71,104)(26,121,72,83)(27,140,73,102)(28,159,74,81)(29,138,75,100)(30,157,76,119)(31,136,77,98)(32,155,78,117)(33,134,79,96)(34,153,80,115)(35,132,41,94)(36,151,42,113)(37,130,43,92)(38,149,44,111)(39,128,45,90)(40,147,46,109), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,73)(42,62)(43,51)(44,80)(45,69)(46,58)(48,76)(49,65)(50,54)(52,72)(53,61)(55,79)(56,68)(59,75)(60,64)(63,71)(66,78)(70,74)(81,143)(82,132)(83,121)(84,150)(85,139)(86,128)(87,157)(88,146)(89,135)(90,124)(91,153)(92,142)(93,131)(94,160)(95,149)(96,138)(97,127)(98,156)(99,145)(100,134)(101,123)(102,152)(103,141)(104,130)(105,159)(106,148)(107,137)(108,126)(109,155)(110,144)(111,133)(112,122)(113,151)(114,140)(115,129)(116,158)(117,147)(118,136)(119,125)(120,154) );
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)], [(1,126,47,88),(2,145,48,107),(3,124,49,86),(4,143,50,105),(5,122,51,84),(6,141,52,103),(7,160,53,82),(8,139,54,101),(9,158,55,120),(10,137,56,99),(11,156,57,118),(12,135,58,97),(13,154,59,116),(14,133,60,95),(15,152,61,114),(16,131,62,93),(17,150,63,112),(18,129,64,91),(19,148,65,110),(20,127,66,89),(21,146,67,108),(22,125,68,87),(23,144,69,106),(24,123,70,85),(25,142,71,104),(26,121,72,83),(27,140,73,102),(28,159,74,81),(29,138,75,100),(30,157,76,119),(31,136,77,98),(32,155,78,117),(33,134,79,96),(34,153,80,115),(35,132,41,94),(36,151,42,113),(37,130,43,92),(38,149,44,111),(39,128,45,90),(40,147,46,109)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,73),(42,62),(43,51),(44,80),(45,69),(46,58),(48,76),(49,65),(50,54),(52,72),(53,61),(55,79),(56,68),(59,75),(60,64),(63,71),(66,78),(70,74),(81,143),(82,132),(83,121),(84,150),(85,139),(86,128),(87,157),(88,146),(89,135),(90,124),(91,153),(92,142),(93,131),(94,160),(95,149),(96,138),(97,127),(98,156),(99,145),(100,134),(101,123),(102,152),(103,141),(104,130),(105,159),(106,148),(107,137),(108,126),(109,155),(110,144),(111,133),(112,122),(113,151),(114,140),(115,129),(116,158),(117,147),(118,136),(119,125),(120,154)]])
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 | 20 | 2 | 2 | 8 | 8 | 20 | 40 | 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 | 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 | C8.C22 | D4⋊2D5 | D4×D5 | Q16⋊D5 |
| kernel | C40.36D4 | C40⋊6C4 | Q8⋊Dic5 | C2×C8⋊D5 | D10⋊3Q8 | C10×Q16 | C40 | C2×Dic5 | C22×D5 | C2×Q16 | C20 | C2×C8 | C2×Q8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.36D4 ►in GL8(𝔽41)
| 1 | 20 | 35 | 0 | 0 | 0 | 0 | 0 |
| 10 | 38 | 35 | 35 | 0 | 0 | 0 | 0 |
| 21 | 0 | 20 | 21 | 0 | 0 | 0 | 0 |
| 18 | 21 | 14 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 | 10 | 5 |
| 0 | 0 | 0 | 0 | 25 | 35 | 25 | 35 |
| 0 | 0 | 0 | 0 | 31 | 36 | 10 | 5 |
| 0 | 0 | 0 | 0 | 16 | 6 | 25 | 35 |
| 6 | 40 | 36 | 1 | 0 | 0 | 0 | 0 |
| 1 | 34 | 7 | 6 | 0 | 0 | 0 | 0 |
| 38 | 21 | 1 | 0 | 0 | 0 | 0 | 0 |
| 31 | 22 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 | 24 | 20 |
| 0 | 0 | 0 | 0 | 40 | 34 | 20 | 17 |
| 0 | 0 | 0 | 0 | 24 | 20 | 34 | 1 |
| 0 | 0 | 0 | 0 | 20 | 17 | 1 | 7 |
| 34 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 20 | 40 | 0 | 0 | 0 | 0 | 0 |
| 9 | 19 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 35 |
G:=sub<GL(8,GF(41))| [1,10,21,18,0,0,0,0,20,38,0,21,0,0,0,0,35,35,20,14,0,0,0,0,0,35,21,23,0,0,0,0,0,0,0,0,10,25,31,16,0,0,0,0,5,35,36,6,0,0,0,0,10,25,10,25,0,0,0,0,5,35,5,35],[6,1,38,31,0,0,0,0,40,34,21,22,0,0,0,0,36,7,1,1,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,7,40,24,20,0,0,0,0,40,34,20,17,0,0,0,0,24,20,34,1,0,0,0,0,20,17,1,7],[34,34,3,9,0,0,0,0,1,7,20,19,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,36,0,0,0,0,0,0,7,35,0,0,0,0,0,0,0,0,6,36,0,0,0,0,0,0,7,35] >;
C40.36D4 in GAP, Magma, Sage, TeX
C_{40}._{36}D_4 % in TeX
G:=Group("C40.36D4"); // GroupNames label
G:=SmallGroup(320,816);
// by ID
G=gap.SmallGroup(320,816);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,219,184,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^29,c*b*c=a^20*b^-1>;
// generators/relations