metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊7D20, C40⋊4D4, D10⋊2D8, C2.D8⋊3D5, C5⋊2(C8⋊7D4), C2.14(D5×D8), (C2×D40)⋊16C2, C4⋊D20⋊7C2, C4⋊C4.48D10, C4.53(C2×D20), C10.30(C2×D8), (C2×C8).231D10, C20.133(C2×D4), D20⋊6C4⋊21C2, C10.75(C4○D8), C20.38(C4○D4), (C2×C40).83C22, C4.9(Q8⋊2D5), (C22×D5).86D4, C22.229(D4×D5), C10.46(C4⋊D4), C2.19(C4⋊D20), (C2×C20).299C23, (C2×Dic5).148D4, (C2×D20).86C22, C2.13(Q8.D10), (D5×C2×C8)⋊2C2, (C5×C2.D8)⋊5C2, (C2×C10).304(C2×D4), (C5×C4⋊C4).92C22, (C2×C4×D5).306C22, (C2×C4).402(C22×D5), (C2×C5⋊2C8).243C22, SmallGroup(320,510)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊7D20
G = < a,b,c | a8=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 742 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C8⋊7D4, C8×D5, D40, C2×C5⋊2C8, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, D20⋊6C4, C5×C2.D8, C4⋊D20, D5×C2×C8, C2×D40, C8⋊7D20
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C4○D8, D20, C22×D5, C8⋊7D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D5×D8, Q8.D10, C8⋊7D20
►On 160 points
Generators in S160
(1 124 46 89 74 111 144 23)(2 24 145 112 75 90 47 125)(3 126 48 91 76 113 146 25)(4 26 147 114 77 92 49 127)(5 128 50 93 78 115 148 27)(6 28 149 116 79 94 51 129)(7 130 52 95 80 117 150 29)(8 30 151 118 61 96 53 131)(9 132 54 97 62 119 152 31)(10 32 153 120 63 98 55 133)(11 134 56 99 64 101 154 33)(12 34 155 102 65 100 57 135)(13 136 58 81 66 103 156 35)(14 36 157 104 67 82 59 137)(15 138 60 83 68 105 158 37)(16 38 159 106 69 84 41 139)(17 140 42 85 70 107 160 39)(18 40 141 108 71 86 43 121)(19 122 44 87 72 109 142 21)(20 22 143 110 73 88 45 123)
(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)(21 140)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 128)(34 127)(35 126)(36 125)(37 124)(38 123)(39 122)(40 121)(41 143)(42 142)(43 141)(44 160)(45 159)(46 158)(47 157)(48 156)(49 155)(50 154)(51 153)(52 152)(53 151)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 113)(82 112)(83 111)(84 110)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)
G:=sub<Sym(160)| (1,124,46,89,74,111,144,23)(2,24,145,112,75,90,47,125)(3,126,48,91,76,113,146,25)(4,26,147,114,77,92,49,127)(5,128,50,93,78,115,148,27)(6,28,149,116,79,94,51,129)(7,130,52,95,80,117,150,29)(8,30,151,118,61,96,53,131)(9,132,54,97,62,119,152,31)(10,32,153,120,63,98,55,133)(11,134,56,99,64,101,154,33)(12,34,155,102,65,100,57,135)(13,136,58,81,66,103,156,35)(14,36,157,104,67,82,59,137)(15,138,60,83,68,105,158,37)(16,38,159,106,69,84,41,139)(17,140,42,85,70,107,160,39)(18,40,141,108,71,86,43,121)(19,122,44,87,72,109,142,21)(20,22,143,110,73,88,45,123), (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)(21,140)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,143)(42,142)(43,141)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)>;
G:=Group( (1,124,46,89,74,111,144,23)(2,24,145,112,75,90,47,125)(3,126,48,91,76,113,146,25)(4,26,147,114,77,92,49,127)(5,128,50,93,78,115,148,27)(6,28,149,116,79,94,51,129)(7,130,52,95,80,117,150,29)(8,30,151,118,61,96,53,131)(9,132,54,97,62,119,152,31)(10,32,153,120,63,98,55,133)(11,134,56,99,64,101,154,33)(12,34,155,102,65,100,57,135)(13,136,58,81,66,103,156,35)(14,36,157,104,67,82,59,137)(15,138,60,83,68,105,158,37)(16,38,159,106,69,84,41,139)(17,140,42,85,70,107,160,39)(18,40,141,108,71,86,43,121)(19,122,44,87,72,109,142,21)(20,22,143,110,73,88,45,123), (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)(21,140)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,143)(42,142)(43,141)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,113)(82,112)(83,111)(84,110)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114) );
G=PermutationGroup([[(1,124,46,89,74,111,144,23),(2,24,145,112,75,90,47,125),(3,126,48,91,76,113,146,25),(4,26,147,114,77,92,49,127),(5,128,50,93,78,115,148,27),(6,28,149,116,79,94,51,129),(7,130,52,95,80,117,150,29),(8,30,151,118,61,96,53,131),(9,132,54,97,62,119,152,31),(10,32,153,120,63,98,55,133),(11,134,56,99,64,101,154,33),(12,34,155,102,65,100,57,135),(13,136,58,81,66,103,156,35),(14,36,157,104,67,82,59,137),(15,138,60,83,68,105,158,37),(16,38,159,106,69,84,41,139),(17,140,42,85,70,107,160,39),(18,40,141,108,71,86,43,121),(19,122,44,87,72,109,142,21),(20,22,143,110,73,88,45,123)], [(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),(21,140),(22,139),(23,138),(24,137),(25,136),(26,135),(27,134),(28,133),(29,132),(30,131),(31,130),(32,129),(33,128),(34,127),(35,126),(36,125),(37,124),(38,123),(39,122),(40,121),(41,143),(42,142),(43,141),(44,160),(45,159),(46,158),(47,157),(48,156),(49,155),(50,154),(51,153),(52,152),(53,151),(54,150),(55,149),(56,148),(57,147),(58,146),(59,145),(60,144),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,113),(82,112),(83,111),(84,110),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 40 | 40 | 2 | 2 | 8 | 8 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D8 | D10 | D10 | C4○D8 | D20 | Q8⋊2D5 | D4×D5 | D5×D8 | Q8.D10 |
| kernel | C8⋊7D20 | D20⋊6C4 | C5×C2.D8 | C4⋊D20 | D5×C2×C8 | C2×D40 | C40 | C2×Dic5 | C22×D5 | C2.D8 | C20 | D10 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C8⋊7D20 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 15 | 17 |
| 32 | 30 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 40 | 9 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,15,0,0,30,17],[32,11,0,0,30,27,0,0,0,0,32,40,0,0,0,9],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,18,1] >;
C8⋊7D20 in GAP, Magma, Sage, TeX
C_8\rtimes_7D_{20} % in TeX
G:=Group("C8:7D20"); // GroupNames label
G:=SmallGroup(320,510);
// by ID
G=gap.SmallGroup(320,510);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations