metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊5(C8×D4), C5⋊D4⋊2C8, D10⋊4(C2×C8), C5⋊2C8⋊29D4, C22⋊1(C8×D5), C22⋊C8⋊16D5, Dic5⋊2(C2×C8), C10.56(C4×D4), C4.194(D4×D5), (C8×Dic5)⋊15C2, C20.353(C2×D4), (C2×C8).194D10, D10⋊1C8⋊16C2, C23.23(C4×D5), C10.48(C8○D4), C10.31(C22×C8), C20.8Q8⋊18C2, C23.D5.11C4, D10⋊C4.19C4, C20.297(C4○D4), (C2×C40).170C22, (C2×C20).820C23, C10.D4.19C4, (C22×C4).303D10, C4.123(D4⋊2D5), C2.3(Dic5⋊4D4), C2.2(D20.2C4), (C22×C20).337C22, (C4×Dic5).302C22, C2.9(D5×C2×C8), (D5×C2×C8)⋊13C2, (C2×C10)⋊6(C2×C8), (C2×C4).63(C4×D5), (C5×C22⋊C8)⋊14C2, C22.44(C2×C4×D5), (C4×C5⋊D4).13C2, (C2×C5⋊D4).12C4, (C2×C20).326(C2×C4), (C22×C5⋊2C8)⋊16C2, (C2×C4×D5).342C22, (C2×Dic5).94(C2×C4), (C22×D5).72(C2×C4), (C2×C4).762(C22×D5), (C22×C10).106(C2×C4), (C2×C10).176(C22×C4), (C2×C5⋊2C8).308C22, SmallGroup(320,352)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊5(C8×D4)
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 398 in 134 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C5⋊2C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8×D4, C8×D5, C2×C5⋊2C8, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D10⋊1C8, C5×C22⋊C8, D5×C2×C8, C22×C5⋊2C8, C4×C5⋊D4, C5⋊5(C8×D4)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D5, C2×C8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C22×C8, C8○D4, C4×D5, C22×D5, C8×D4, C8×D5, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, D5×C2×C8, D20.2C4, C5⋊5(C8×D4)
►On 160 points
Generators in S160
(1 145 114 34 99)(2 100 35 115 146)(3 147 116 36 101)(4 102 37 117 148)(5 149 118 38 103)(6 104 39 119 150)(7 151 120 40 97)(8 98 33 113 152)(9 92 124 63 41)(10 42 64 125 93)(11 94 126 57 43)(12 44 58 127 95)(13 96 128 59 45)(14 46 60 121 89)(15 90 122 61 47)(16 48 62 123 91)(17 28 56 139 136)(18 129 140 49 29)(19 30 50 141 130)(20 131 142 51 31)(21 32 52 143 132)(22 133 144 53 25)(23 26 54 137 134)(24 135 138 55 27)(65 157 106 88 80)(66 73 81 107 158)(67 159 108 82 74)(68 75 83 109 160)(69 153 110 84 76)(70 77 85 111 154)(71 155 112 86 78)(72 79 87 105 156)
(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)
(1 111 48 25)(2 112 41 26)(3 105 42 27)(4 106 43 28)(5 107 44 29)(6 108 45 30)(7 109 46 31)(8 110 47 32)(9 23 100 155)(10 24 101 156)(11 17 102 157)(12 18 103 158)(13 19 104 159)(14 20 97 160)(15 21 98 153)(16 22 99 154)(33 69 90 132)(34 70 91 133)(35 71 92 134)(36 72 93 135)(37 65 94 136)(38 66 95 129)(39 67 96 130)(40 68 89 131)(49 149 81 58)(50 150 82 59)(51 151 83 60)(52 152 84 61)(53 145 85 62)(54 146 86 63)(55 147 87 64)(56 148 88 57)(73 127 140 118)(74 128 141 119)(75 121 142 120)(76 122 143 113)(77 123 144 114)(78 124 137 115)(79 125 138 116)(80 126 139 117)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 81)(17 152)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(33 139)(34 140)(35 141)(36 142)(37 143)(38 144)(39 137)(40 138)(41 108)(42 109)(43 110)(44 111)(45 112)(46 105)(47 106)(48 107)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 121)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 89)(80 90)(113 136)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)
G:=sub<Sym(160)| (1,145,114,34,99)(2,100,35,115,146)(3,147,116,36,101)(4,102,37,117,148)(5,149,118,38,103)(6,104,39,119,150)(7,151,120,40,97)(8,98,33,113,152)(9,92,124,63,41)(10,42,64,125,93)(11,94,126,57,43)(12,44,58,127,95)(13,96,128,59,45)(14,46,60,121,89)(15,90,122,61,47)(16,48,62,123,91)(17,28,56,139,136)(18,129,140,49,29)(19,30,50,141,130)(20,131,142,51,31)(21,32,52,143,132)(22,133,144,53,25)(23,26,54,137,134)(24,135,138,55,27)(65,157,106,88,80)(66,73,81,107,158)(67,159,108,82,74)(68,75,83,109,160)(69,153,110,84,76)(70,77,85,111,154)(71,155,112,86,78)(72,79,87,105,156), (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), (1,111,48,25)(2,112,41,26)(3,105,42,27)(4,106,43,28)(5,107,44,29)(6,108,45,30)(7,109,46,31)(8,110,47,32)(9,23,100,155)(10,24,101,156)(11,17,102,157)(12,18,103,158)(13,19,104,159)(14,20,97,160)(15,21,98,153)(16,22,99,154)(33,69,90,132)(34,70,91,133)(35,71,92,134)(36,72,93,135)(37,65,94,136)(38,66,95,129)(39,67,96,130)(40,68,89,131)(49,149,81,58)(50,150,82,59)(51,151,83,60)(52,152,84,61)(53,145,85,62)(54,146,86,63)(55,147,87,64)(56,148,88,57)(73,127,140,118)(74,128,141,119)(75,121,142,120)(76,122,143,113)(77,123,144,114)(78,124,137,115)(79,125,138,116)(80,126,139,117), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,81)(17,152)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(33,139)(34,140)(35,141)(36,142)(37,143)(38,144)(39,137)(40,138)(41,108)(42,109)(43,110)(44,111)(45,112)(46,105)(47,106)(48,107)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,97)(56,98)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,121)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,89)(80,90)(113,136)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)>;
G:=Group( (1,145,114,34,99)(2,100,35,115,146)(3,147,116,36,101)(4,102,37,117,148)(5,149,118,38,103)(6,104,39,119,150)(7,151,120,40,97)(8,98,33,113,152)(9,92,124,63,41)(10,42,64,125,93)(11,94,126,57,43)(12,44,58,127,95)(13,96,128,59,45)(14,46,60,121,89)(15,90,122,61,47)(16,48,62,123,91)(17,28,56,139,136)(18,129,140,49,29)(19,30,50,141,130)(20,131,142,51,31)(21,32,52,143,132)(22,133,144,53,25)(23,26,54,137,134)(24,135,138,55,27)(65,157,106,88,80)(66,73,81,107,158)(67,159,108,82,74)(68,75,83,109,160)(69,153,110,84,76)(70,77,85,111,154)(71,155,112,86,78)(72,79,87,105,156), (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), (1,111,48,25)(2,112,41,26)(3,105,42,27)(4,106,43,28)(5,107,44,29)(6,108,45,30)(7,109,46,31)(8,110,47,32)(9,23,100,155)(10,24,101,156)(11,17,102,157)(12,18,103,158)(13,19,104,159)(14,20,97,160)(15,21,98,153)(16,22,99,154)(33,69,90,132)(34,70,91,133)(35,71,92,134)(36,72,93,135)(37,65,94,136)(38,66,95,129)(39,67,96,130)(40,68,89,131)(49,149,81,58)(50,150,82,59)(51,151,83,60)(52,152,84,61)(53,145,85,62)(54,146,86,63)(55,147,87,64)(56,148,88,57)(73,127,140,118)(74,128,141,119)(75,121,142,120)(76,122,143,113)(77,123,144,114)(78,124,137,115)(79,125,138,116)(80,126,139,117), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,81)(17,152)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(33,139)(34,140)(35,141)(36,142)(37,143)(38,144)(39,137)(40,138)(41,108)(42,109)(43,110)(44,111)(45,112)(46,105)(47,106)(48,107)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,97)(56,98)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,121)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,89)(80,90)(113,136)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135) );
G=PermutationGroup([[(1,145,114,34,99),(2,100,35,115,146),(3,147,116,36,101),(4,102,37,117,148),(5,149,118,38,103),(6,104,39,119,150),(7,151,120,40,97),(8,98,33,113,152),(9,92,124,63,41),(10,42,64,125,93),(11,94,126,57,43),(12,44,58,127,95),(13,96,128,59,45),(14,46,60,121,89),(15,90,122,61,47),(16,48,62,123,91),(17,28,56,139,136),(18,129,140,49,29),(19,30,50,141,130),(20,131,142,51,31),(21,32,52,143,132),(22,133,144,53,25),(23,26,54,137,134),(24,135,138,55,27),(65,157,106,88,80),(66,73,81,107,158),(67,159,108,82,74),(68,75,83,109,160),(69,153,110,84,76),(70,77,85,111,154),(71,155,112,86,78),(72,79,87,105,156)], [(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)], [(1,111,48,25),(2,112,41,26),(3,105,42,27),(4,106,43,28),(5,107,44,29),(6,108,45,30),(7,109,46,31),(8,110,47,32),(9,23,100,155),(10,24,101,156),(11,17,102,157),(12,18,103,158),(13,19,104,159),(14,20,97,160),(15,21,98,153),(16,22,99,154),(33,69,90,132),(34,70,91,133),(35,71,92,134),(36,72,93,135),(37,65,94,136),(38,66,95,129),(39,67,96,130),(40,68,89,131),(49,149,81,58),(50,150,82,59),(51,151,83,60),(52,152,84,61),(53,145,85,62),(54,146,86,63),(55,147,87,64),(56,148,88,57),(73,127,140,118),(74,128,141,119),(75,121,142,120),(76,122,143,113),(77,123,144,114),(78,124,137,115),(79,125,138,116),(80,126,139,117)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,82),(10,83),(11,84),(12,85),(13,86),(14,87),(15,88),(16,81),(17,152),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(33,139),(34,140),(35,141),(36,142),(37,143),(38,144),(39,137),(40,138),(41,108),(42,109),(43,110),(44,111),(45,112),(46,105),(47,106),(48,107),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,97),(56,98),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,121),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,89),(80,90),(113,136),(114,129),(115,130),(116,131),(117,132),(118,133),(119,134),(120,135)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 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 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C4 | C4 | C4 | C4 | C8 | D4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | C8×D5 | D4×D5 | D4⋊2D5 | D20.2C4 |
| kernel | C5⋊5(C8×D4) | C8×Dic5 | C20.8Q8 | D10⋊1C8 | C5×C22⋊C8 | D5×C2×C8 | C22×C5⋊2C8 | C4×C5⋊D4 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C5⋊D4 | C5⋊2C8 | C22⋊C8 | C20 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C5⋊5(C8×D4) ►in GL4(𝔽41) generated by
| 34 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 38 | 21 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 1 | 34 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 18 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 32 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,21,3,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,34,40,0,0,0,0,1,18,0,0,9,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,32,1] >;
C5⋊5(C8×D4) in GAP, Magma, Sage, TeX
C_5\rtimes_5(C_8\times D_4)
% in TeX
G:=Group("C5:5(C8xD4)"); // GroupNames label
G:=SmallGroup(320,352);
// by ID
G=gap.SmallGroup(320,352);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,219,58,136,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations