metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6D20, C40⋊25D4, C20⋊10M4(2), C42.257D10, (C4×C8)⋊14D5, (C4×C40)⋊21C2, C5⋊4(C8⋊6D4), C20⋊3C8⋊2C2, C2.8(C4×D20), C4⋊2(C8⋊D5), (C4×D20).2C2, C4.75(C2×D20), C10.35(C4×D4), D10⋊1C8⋊1C2, (C2×D20).17C4, C20.295(C2×D4), (C2×C8).283D10, C4⋊Dic5.19C4, C10.28(C8○D4), D10⋊C4.11C4, C4.126(C4○D20), C20.242(C4○D4), (C4×C20).324C22, (C2×C40).343C22, (C2×C20).806C23, C10.38(C2×M4(2)), C2.7(D20.3C4), C2.7(C2×C8⋊D5), (C2×C8⋊D5)⋊10C2, C22.95(C2×C4×D5), (C2×C4).104(C4×D5), (C2×C20).394(C2×C4), (C2×C4×D5).226C22, (C2×Dic5).13(C2×C4), (C22×D5).14(C2×C4), (C2×C4).748(C22×D5), (C2×C10).162(C22×C4), (C2×C5⋊2C8).191C22, SmallGroup(320,315)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊6D20
G = < a,b,c | a8=b20=c2=1, ab=ba, cac=a5, cbc=b-1 >
Subgroups: 446 in 122 conjugacy classes, 55 normal (33 characteristic)
C1, C2, C2, C4, 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, C20, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊2C8, C40, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C8⋊6D4, C8⋊D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C20⋊3C8, D10⋊1C8, C4×C40, C4×D20, C2×C8⋊D5, C8⋊6D20
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⋊6D4, C8⋊D5, C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×C8⋊D5, D20.3C4, C8⋊6D20
►On 160 points
Generators in S160
(1 107 50 96 123 38 144 61)(2 108 51 97 124 39 145 62)(3 109 52 98 125 40 146 63)(4 110 53 99 126 21 147 64)(5 111 54 100 127 22 148 65)(6 112 55 81 128 23 149 66)(7 113 56 82 129 24 150 67)(8 114 57 83 130 25 151 68)(9 115 58 84 131 26 152 69)(10 116 59 85 132 27 153 70)(11 117 60 86 133 28 154 71)(12 118 41 87 134 29 155 72)(13 119 42 88 135 30 156 73)(14 120 43 89 136 31 157 74)(15 101 44 90 137 32 158 75)(16 102 45 91 138 33 159 76)(17 103 46 92 139 34 160 77)(18 104 47 93 140 35 141 78)(19 105 48 94 121 36 142 79)(20 106 49 95 122 37 143 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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 103)(22 102)(23 101)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 104)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(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)(121 124)(122 123)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(141 146)(142 145)(143 144)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)
G:=sub<Sym(160)| (1,107,50,96,123,38,144,61)(2,108,51,97,124,39,145,62)(3,109,52,98,125,40,146,63)(4,110,53,99,126,21,147,64)(5,111,54,100,127,22,148,65)(6,112,55,81,128,23,149,66)(7,113,56,82,129,24,150,67)(8,114,57,83,130,25,151,68)(9,115,58,84,131,26,152,69)(10,116,59,85,132,27,153,70)(11,117,60,86,133,28,154,71)(12,118,41,87,134,29,155,72)(13,119,42,88,135,30,156,73)(14,120,43,89,136,31,157,74)(15,101,44,90,137,32,158,75)(16,102,45,91,138,33,159,76)(17,103,46,92,139,34,160,77)(18,104,47,93,140,35,141,78)(19,105,48,94,121,36,142,79)(20,106,49,95,122,37,143,80), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,103)(22,102)(23,101)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(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)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,146)(142,145)(143,144)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)>;
G:=Group( (1,107,50,96,123,38,144,61)(2,108,51,97,124,39,145,62)(3,109,52,98,125,40,146,63)(4,110,53,99,126,21,147,64)(5,111,54,100,127,22,148,65)(6,112,55,81,128,23,149,66)(7,113,56,82,129,24,150,67)(8,114,57,83,130,25,151,68)(9,115,58,84,131,26,152,69)(10,116,59,85,132,27,153,70)(11,117,60,86,133,28,154,71)(12,118,41,87,134,29,155,72)(13,119,42,88,135,30,156,73)(14,120,43,89,136,31,157,74)(15,101,44,90,137,32,158,75)(16,102,45,91,138,33,159,76)(17,103,46,92,139,34,160,77)(18,104,47,93,140,35,141,78)(19,105,48,94,121,36,142,79)(20,106,49,95,122,37,143,80), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,103)(22,102)(23,101)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(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)(121,124)(122,123)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)(141,146)(142,145)(143,144)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154) );
G=PermutationGroup([[(1,107,50,96,123,38,144,61),(2,108,51,97,124,39,145,62),(3,109,52,98,125,40,146,63),(4,110,53,99,126,21,147,64),(5,111,54,100,127,22,148,65),(6,112,55,81,128,23,149,66),(7,113,56,82,129,24,150,67),(8,114,57,83,130,25,151,68),(9,115,58,84,131,26,152,69),(10,116,59,85,132,27,153,70),(11,117,60,86,133,28,154,71),(12,118,41,87,134,29,155,72),(13,119,42,88,135,30,156,73),(14,120,43,89,136,31,157,74),(15,101,44,90,137,32,158,75),(16,102,45,91,138,33,159,76),(17,103,46,92,139,34,160,77),(18,104,47,93,140,35,141,78),(19,105,48,94,121,36,142,79),(20,106,49,95,122,37,143,80)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,103),(22,102),(23,101),(24,120),(25,119),(26,118),(27,117),(28,116),(29,115),(30,114),(31,113),(32,112),(33,111),(34,110),(35,109),(36,108),(37,107),(38,106),(39,105),(40,104),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(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),(121,124),(122,123),(125,140),(126,139),(127,138),(128,137),(129,136),(130,135),(131,134),(132,133),(141,146),(142,145),(143,144),(147,160),(148,159),(149,158),(150,157),(151,156),(152,155),(153,154)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | D20 | C4×D5 | C8⋊D5 | C4○D20 | D20.3C4 |
| kernel | C8⋊6D20 | C20⋊3C8 | D10⋊1C8 | C4×C40 | C4×D20 | C2×C8⋊D5 | C4⋊Dic5 | D10⋊C4 | C2×D20 | C40 | C4×C8 | C20 | C20 | C42 | C2×C8 | C10 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 8 | 16 |
Matrix representation of C8⋊6D20 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 35 | 2 |
| 0 | 0 | 39 | 6 |
| 27 | 30 | 0 | 0 |
| 11 | 32 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 35 |
| 11 | 14 | 0 | 0 |
| 9 | 30 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,35,39,0,0,2,6],[27,11,0,0,30,32,0,0,0,0,0,1,0,0,40,35],[11,9,0,0,14,30,0,0,0,0,0,40,0,0,40,0] >;
C8⋊6D20 in GAP, Magma, Sage, TeX
C_8\rtimes_6D_{20} % in TeX
G:=Group("C8:6D20"); // GroupNames label
G:=SmallGroup(320,315);
// by ID
G=gap.SmallGroup(320,315);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations