metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊9D20, C40⋊16D4, D10⋊3M4(2), C42.15D10, C8⋊C4⋊8D5, C5⋊5(C8⋊9D4), C20⋊3C8⋊3C2, (C4×D20).4C2, C10.40(C4×D4), C2.13(C4×D20), C4.77(C2×D20), (C2×D20).18C4, C20.297(C2×D4), (C2×C8).157D10, C4⋊Dic5.21C4, D10⋊1C8⋊36C2, C10.45(C8○D4), (C4×C20).13C22, C2.10(D5×M4(2)), D10⋊C4.14C4, C4.130(C4○D20), C20.246(C4○D4), (C2×C20).812C23, (C2×C40).226C22, C10.51(C2×M4(2)), C2.7(D20.2C4), (D5×C2×C8)⋊26C2, (C5×C8⋊C4)⋊7C2, (C2×C4).29(C4×D5), (C2×C8⋊D5)⋊24C2, C22.99(C2×C4×D5), (C2×C20).210(C2×C4), (C2×C4×D5).228C22, (C2×Dic5).16(C2×C4), (C22×D5).70(C2×C4), (C2×C4).754(C22×D5), (C2×C10).168(C22×C4), (C2×C5⋊2C8).303C22, SmallGroup(320,333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊9D20
G = < a,b,c | a8=b20=c2=1, bab-1=cac=a5, cbc=b-1 >
Subgroups: 446 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C40, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C8⋊9D4, C8×D5, C8⋊D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C20⋊3C8, D10⋊1C8, C5×C8⋊C4, C4×D20, D5×C2×C8, C2×C8⋊D5, C8⋊9D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, D20, C22×D5, C8⋊9D4, C2×C4×D5, C2×D20, C4○D20, C4×D20, D5×M4(2), D20.2C4, C8⋊9D20
►On 160 points
Generators in S160
(1 94 23 138 141 68 58 104)(2 69 24 105 142 95 59 139)(3 96 25 140 143 70 60 106)(4 71 26 107 144 97 41 121)(5 98 27 122 145 72 42 108)(6 73 28 109 146 99 43 123)(7 100 29 124 147 74 44 110)(8 75 30 111 148 81 45 125)(9 82 31 126 149 76 46 112)(10 77 32 113 150 83 47 127)(11 84 33 128 151 78 48 114)(12 79 34 115 152 85 49 129)(13 86 35 130 153 80 50 116)(14 61 36 117 154 87 51 131)(15 88 37 132 155 62 52 118)(16 63 38 119 156 89 53 133)(17 90 39 134 157 64 54 120)(18 65 40 101 158 91 55 135)(19 92 21 136 159 66 56 102)(20 67 22 103 160 93 57 137)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 100)(77 99)(78 98)(79 97)(80 96)(101 135)(102 134)(103 133)(104 132)(105 131)(106 130)(107 129)(108 128)(109 127)(110 126)(111 125)(112 124)(113 123)(114 122)(115 121)(116 140)(117 139)(118 138)(119 137)(120 136)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
G:=sub<Sym(160)| (1,94,23,138,141,68,58,104)(2,69,24,105,142,95,59,139)(3,96,25,140,143,70,60,106)(4,71,26,107,144,97,41,121)(5,98,27,122,145,72,42,108)(6,73,28,109,146,99,43,123)(7,100,29,124,147,74,44,110)(8,75,30,111,148,81,45,125)(9,82,31,126,149,76,46,112)(10,77,32,113,150,83,47,127)(11,84,33,128,151,78,48,114)(12,79,34,115,152,85,49,129)(13,86,35,130,153,80,50,116)(14,61,36,117,154,87,51,131)(15,88,37,132,155,62,52,118)(16,63,38,119,156,89,53,133)(17,90,39,134,157,64,54,120)(18,65,40,101,158,91,55,135)(19,92,21,136,159,66,56,102)(20,67,22,103,160,93,57,137), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,100)(77,99)(78,98)(79,97)(80,96)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,140)(117,139)(118,138)(119,137)(120,136)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159)>;
G:=Group( (1,94,23,138,141,68,58,104)(2,69,24,105,142,95,59,139)(3,96,25,140,143,70,60,106)(4,71,26,107,144,97,41,121)(5,98,27,122,145,72,42,108)(6,73,28,109,146,99,43,123)(7,100,29,124,147,74,44,110)(8,75,30,111,148,81,45,125)(9,82,31,126,149,76,46,112)(10,77,32,113,150,83,47,127)(11,84,33,128,151,78,48,114)(12,79,34,115,152,85,49,129)(13,86,35,130,153,80,50,116)(14,61,36,117,154,87,51,131)(15,88,37,132,155,62,52,118)(16,63,38,119,156,89,53,133)(17,90,39,134,157,64,54,120)(18,65,40,101,158,91,55,135)(19,92,21,136,159,66,56,102)(20,67,22,103,160,93,57,137), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,100)(77,99)(78,98)(79,97)(80,96)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,140)(117,139)(118,138)(119,137)(120,136)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159) );
G=PermutationGroup([[(1,94,23,138,141,68,58,104),(2,69,24,105,142,95,59,139),(3,96,25,140,143,70,60,106),(4,71,26,107,144,97,41,121),(5,98,27,122,145,72,42,108),(6,73,28,109,146,99,43,123),(7,100,29,124,147,74,44,110),(8,75,30,111,148,81,45,125),(9,82,31,126,149,76,46,112),(10,77,32,113,150,83,47,127),(11,84,33,128,151,78,48,114),(12,79,34,115,152,85,49,129),(13,86,35,130,153,80,50,116),(14,61,36,117,154,87,51,131),(15,88,37,132,155,62,52,118),(16,63,38,119,156,89,53,133),(17,90,39,134,157,64,54,120),(18,65,40,101,158,91,55,135),(19,92,21,136,159,66,56,102),(20,67,22,103,160,93,57,137)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,100),(77,99),(78,98),(79,97),(80,96),(101,135),(102,134),(103,133),(104,132),(105,131),(106,130),(107,129),(108,128),(109,127),(110,126),(111,125),(112,124),(113,123),(114,122),(115,121),(116,140),(117,139),(118,138),(119,137),(120,136),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4○D4 | M4(2) | D10 | D10 | C8○D4 | D20 | C4×D5 | C4○D20 | D5×M4(2) | D20.2C4 |
| kernel | C8⋊9D20 | C20⋊3C8 | D10⋊1C8 | C5×C8⋊C4 | C4×D20 | D5×C2×C8 | C2×C8⋊D5 | C4⋊Dic5 | D10⋊C4 | C2×D20 | C40 | C8⋊C4 | C20 | D10 | C42 | C2×C8 | C10 | C8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C8⋊9D20 ►in GL4(𝔽41) generated by
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 21 |
| 0 | 0 | 21 | 1 |
| 16 | 30 | 0 | 0 |
| 27 | 2 | 0 | 0 |
| 0 | 0 | 3 | 19 |
| 0 | 0 | 34 | 38 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 37 | 1 |
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,40,21,0,0,21,1],[16,27,0,0,30,2,0,0,0,0,3,34,0,0,19,38],[1,0,0,0,1,40,0,0,0,0,40,37,0,0,0,1] >;
C8⋊9D20 in GAP, Magma, Sage, TeX
C_8\rtimes_9D_{20} % in TeX
G:=Group("C8:9D20"); // GroupNames label
G:=SmallGroup(320,333);
// by ID
G=gap.SmallGroup(320,333);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^-1>;
// generators/relations