metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8⋊24D4, C5⋊5(C8⋊8D4), C22⋊Q8⋊3D5, C4⋊C4.67D10, (C2×C10)⋊6SD16, C4.174(D4×D5), (C2×C20).267D4, C20.154(C2×D4), C22⋊1(Q8⋊D5), (C2×Q8).29D10, D20⋊6C4⋊39C2, C20⋊7D4.12C2, Q8⋊Dic5⋊15C2, C20.Q8⋊39C2, C10.73(C2×SD16), (C22×C10).94D4, C20.190(C4○D4), C10.101(C4○D8), C4.63(D4⋊2D5), C10.97(C4⋊D4), (C2×C20).367C23, (C22×C4).342D10, C23.41(C5⋊D4), (Q8×C10).47C22, (C2×D20).103C22, C4⋊Dic5.146C22, C2.18(Dic5⋊D4), C2.20(D4.8D10), (C22×C20).171C22, (C2×Q8⋊D5)⋊10C2, C2.10(C2×Q8⋊D5), (C5×C22⋊Q8)⋊3C2, (C22×C5⋊2C8)⋊5C2, (C2×C10).498(C2×D4), (C2×C4).107(C5⋊D4), (C5×C4⋊C4).114C22, (C2×C4).467(C22×D5), C22.173(C2×C5⋊D4), (C2×C5⋊2C8).258C22, SmallGroup(320,675)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊2C8⋊24D4
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, bd=db, dcd=c-1 >
Subgroups: 502 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C5⋊2C8, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C8⋊8D4, C2×C5⋊2C8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, Q8⋊D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, Q8×C10, C20.Q8, D20⋊6C4, Q8⋊Dic5, C22×C5⋊2C8, C20⋊7D4, C2×Q8⋊D5, C5×C22⋊Q8, C5⋊2C8⋊24D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, C5⋊D4, C22×D5, C8⋊8D4, Q8⋊D5, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊D4, C2×Q8⋊D5, D4.8D10, C5⋊2C8⋊24D4
►On 160 points
Generators in S160
(1 99 125 46 145)(2 146 47 126 100)(3 101 127 48 147)(4 148 41 128 102)(5 103 121 42 149)(6 150 43 122 104)(7 97 123 44 151)(8 152 45 124 98)(9 34 63 117 96)(10 89 118 64 35)(11 36 57 119 90)(12 91 120 58 37)(13 38 59 113 92)(14 93 114 60 39)(15 40 61 115 94)(16 95 116 62 33)(17 137 106 56 65)(18 66 49 107 138)(19 139 108 50 67)(20 68 51 109 140)(21 141 110 52 69)(22 70 53 111 142)(23 143 112 54 71)(24 72 55 105 144)(25 73 156 129 87)(26 88 130 157 74)(27 75 158 131 81)(28 82 132 159 76)(29 77 160 133 83)(30 84 134 153 78)(31 79 154 135 85)(32 86 136 155 80)
(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 70 37 131)(2 65 38 134)(3 68 39 129)(4 71 40 132)(5 66 33 135)(6 69 34 130)(7 72 35 133)(8 67 36 136)(9 157 104 21)(10 160 97 24)(11 155 98 19)(12 158 99 22)(13 153 100 17)(14 156 101 20)(15 159 102 23)(16 154 103 18)(25 48 109 114)(26 43 110 117)(27 46 111 120)(28 41 112 115)(29 44 105 118)(30 47 106 113)(31 42 107 116)(32 45 108 119)(49 62 85 149)(50 57 86 152)(51 60 87 147)(52 63 88 150)(53 58 81 145)(54 61 82 148)(55 64 83 151)(56 59 84 146)(73 127 140 93)(74 122 141 96)(75 125 142 91)(76 128 143 94)(77 123 144 89)(78 126 137 92)(79 121 138 95)(80 124 139 90)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(17 148)(18 149)(19 150)(20 151)(21 152)(22 145)(23 146)(24 147)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 135)(34 136)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 103)(50 104)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 157)(58 158)(59 159)(60 160)(61 153)(62 154)(63 155)(64 156)(73 118)(74 119)(75 120)(76 113)(77 114)(78 115)(79 116)(80 117)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)
G:=sub<Sym(160)| (1,99,125,46,145)(2,146,47,126,100)(3,101,127,48,147)(4,148,41,128,102)(5,103,121,42,149)(6,150,43,122,104)(7,97,123,44,151)(8,152,45,124,98)(9,34,63,117,96)(10,89,118,64,35)(11,36,57,119,90)(12,91,120,58,37)(13,38,59,113,92)(14,93,114,60,39)(15,40,61,115,94)(16,95,116,62,33)(17,137,106,56,65)(18,66,49,107,138)(19,139,108,50,67)(20,68,51,109,140)(21,141,110,52,69)(22,70,53,111,142)(23,143,112,54,71)(24,72,55,105,144)(25,73,156,129,87)(26,88,130,157,74)(27,75,158,131,81)(28,82,132,159,76)(29,77,160,133,83)(30,84,134,153,78)(31,79,154,135,85)(32,86,136,155,80), (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), (1,70,37,131)(2,65,38,134)(3,68,39,129)(4,71,40,132)(5,66,33,135)(6,69,34,130)(7,72,35,133)(8,67,36,136)(9,157,104,21)(10,160,97,24)(11,155,98,19)(12,158,99,22)(13,153,100,17)(14,156,101,20)(15,159,102,23)(16,154,103,18)(25,48,109,114)(26,43,110,117)(27,46,111,120)(28,41,112,115)(29,44,105,118)(30,47,106,113)(31,42,107,116)(32,45,108,119)(49,62,85,149)(50,57,86,152)(51,60,87,147)(52,63,88,150)(53,58,81,145)(54,61,82,148)(55,64,83,151)(56,59,84,146)(73,127,140,93)(74,122,141,96)(75,125,142,91)(76,128,143,94)(77,123,144,89)(78,126,137,92)(79,121,138,95)(80,124,139,90), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,86)(10,87)(11,88)(12,81)(13,82)(14,83)(15,84)(16,85)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,135)(34,136)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,103)(50,104)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(73,118)(74,119)(75,120)(76,113)(77,114)(78,115)(79,116)(80,117)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)>;
G:=Group( (1,99,125,46,145)(2,146,47,126,100)(3,101,127,48,147)(4,148,41,128,102)(5,103,121,42,149)(6,150,43,122,104)(7,97,123,44,151)(8,152,45,124,98)(9,34,63,117,96)(10,89,118,64,35)(11,36,57,119,90)(12,91,120,58,37)(13,38,59,113,92)(14,93,114,60,39)(15,40,61,115,94)(16,95,116,62,33)(17,137,106,56,65)(18,66,49,107,138)(19,139,108,50,67)(20,68,51,109,140)(21,141,110,52,69)(22,70,53,111,142)(23,143,112,54,71)(24,72,55,105,144)(25,73,156,129,87)(26,88,130,157,74)(27,75,158,131,81)(28,82,132,159,76)(29,77,160,133,83)(30,84,134,153,78)(31,79,154,135,85)(32,86,136,155,80), (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), (1,70,37,131)(2,65,38,134)(3,68,39,129)(4,71,40,132)(5,66,33,135)(6,69,34,130)(7,72,35,133)(8,67,36,136)(9,157,104,21)(10,160,97,24)(11,155,98,19)(12,158,99,22)(13,153,100,17)(14,156,101,20)(15,159,102,23)(16,154,103,18)(25,48,109,114)(26,43,110,117)(27,46,111,120)(28,41,112,115)(29,44,105,118)(30,47,106,113)(31,42,107,116)(32,45,108,119)(49,62,85,149)(50,57,86,152)(51,60,87,147)(52,63,88,150)(53,58,81,145)(54,61,82,148)(55,64,83,151)(56,59,84,146)(73,127,140,93)(74,122,141,96)(75,125,142,91)(76,128,143,94)(77,123,144,89)(78,126,137,92)(79,121,138,95)(80,124,139,90), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,86)(10,87)(11,88)(12,81)(13,82)(14,83)(15,84)(16,85)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,135)(34,136)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,103)(50,104)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,157)(58,158)(59,159)(60,160)(61,153)(62,154)(63,155)(64,156)(73,118)(74,119)(75,120)(76,113)(77,114)(78,115)(79,116)(80,117)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126) );
G=PermutationGroup([[(1,99,125,46,145),(2,146,47,126,100),(3,101,127,48,147),(4,148,41,128,102),(5,103,121,42,149),(6,150,43,122,104),(7,97,123,44,151),(8,152,45,124,98),(9,34,63,117,96),(10,89,118,64,35),(11,36,57,119,90),(12,91,120,58,37),(13,38,59,113,92),(14,93,114,60,39),(15,40,61,115,94),(16,95,116,62,33),(17,137,106,56,65),(18,66,49,107,138),(19,139,108,50,67),(20,68,51,109,140),(21,141,110,52,69),(22,70,53,111,142),(23,143,112,54,71),(24,72,55,105,144),(25,73,156,129,87),(26,88,130,157,74),(27,75,158,131,81),(28,82,132,159,76),(29,77,160,133,83),(30,84,134,153,78),(31,79,154,135,85),(32,86,136,155,80)], [(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)], [(1,70,37,131),(2,65,38,134),(3,68,39,129),(4,71,40,132),(5,66,33,135),(6,69,34,130),(7,72,35,133),(8,67,36,136),(9,157,104,21),(10,160,97,24),(11,155,98,19),(12,158,99,22),(13,153,100,17),(14,156,101,20),(15,159,102,23),(16,154,103,18),(25,48,109,114),(26,43,110,117),(27,46,111,120),(28,41,112,115),(29,44,105,118),(30,47,106,113),(31,42,107,116),(32,45,108,119),(49,62,85,149),(50,57,86,152),(51,60,87,147),(52,63,88,150),(53,58,81,145),(54,61,82,148),(55,64,83,151),(56,59,84,146),(73,127,140,93),(74,122,141,96),(75,125,142,91),(76,128,143,94),(77,123,144,89),(78,126,137,92),(79,121,138,95),(80,124,139,90)], [(1,70),(2,71),(3,72),(4,65),(5,66),(6,67),(7,68),(8,69),(9,86),(10,87),(11,88),(12,81),(13,82),(14,83),(15,84),(16,85),(17,148),(18,149),(19,150),(20,151),(21,152),(22,145),(23,146),(24,147),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,135),(34,136),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,103),(50,104),(51,97),(52,98),(53,99),(54,100),(55,101),(56,102),(57,157),(58,158),(59,159),(60,160),(61,153),(62,154),(63,155),(64,156),(73,118),(74,119),(75,120),(76,113),(77,114),(78,115),(79,116),(80,117),(105,127),(106,128),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 2 | 2 | 8 | 8 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 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 | C4○D4 | SD16 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4×D5 | D4⋊2D5 | Q8⋊D5 | D4.8D10 |
| kernel | C5⋊2C8⋊24D4 | C20.Q8 | D20⋊6C4 | Q8⋊Dic5 | C22×C5⋊2C8 | C20⋊7D4 | C2×Q8⋊D5 | C5×C22⋊Q8 | C5⋊2C8 | C2×C20 | C22×C10 | C22⋊Q8 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C2×C4 | C23 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C5⋊2C8⋊24D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 6 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 2 | 0 | 0 |
| 0 | 0 | 21 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 |
| 0 | 0 | 0 | 0 | 18 | 20 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 23 | 0 | 0 |
| 0 | 0 | 7 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 18 | 0 | 0 |
| 0 | 0 | 25 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6],[26,26,0,0,0,0,15,26,0,0,0,0,0,0,30,21,0,0,0,0,2,11,0,0,0,0,0,0,21,18,0,0,0,0,21,20],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,17,7,0,0,0,0,23,24,0,0,0,0,0,0,6,40,0,0,0,0,35,35],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,24,25,0,0,0,0,18,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C5⋊2C8⋊24D4 in GAP, Magma, Sage, TeX
C_5\rtimes_2C_8\rtimes_{24}D_4 % in TeX
G:=Group("C5:2C8:24D4"); // GroupNames label
G:=SmallGroup(320,675);
// by ID
G=gap.SmallGroup(320,675);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations