metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊8D20, C40⋊13D4, D10⋊3SD16, C4.Q8⋊8D5, C5⋊3(C8⋊8D4), C4⋊C4.38D10, C4.50(C2×D20), D10⋊2Q8⋊6C2, (C2×C8).260D10, C4⋊D20.6C2, C20.130(C2×D4), D20⋊6C4⋊15C2, C2.24(D5×SD16), C10.55(C4○D8), C20.29(C4○D4), C10.Q16⋊16C2, C4.3(Q8⋊2D5), C10.40(C2×SD16), (C22×D5).84D4, C22.216(D4×D5), C10.43(C4⋊D4), C2.16(C4⋊D20), (C2×C40).161C22, (C2×C20).280C23, (C2×Dic5).144D4, (C2×D20).78C22, C2.22(SD16⋊3D5), (C2×Dic10).87C22, (D5×C2×C8)⋊7C2, (C5×C4.Q8)⋊9C2, (C2×C40⋊C2)⋊28C2, (C2×C10).285(C2×D4), (C5×C4⋊C4).73C22, (C2×C4×D5).303C22, (C2×C4).383(C22×D5), (C2×C5⋊2C8).236C22, SmallGroup(320,491)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊8D20
G = < a,b,c | a8=b20=c2=1, bab-1=cac=a3, cbc=b-1 >
Subgroups: 598 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C8, C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8 [×3], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C5⋊2C8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C8⋊8D4, C8×D5 [×2], C40⋊C2 [×2], C2×C5⋊2C8, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D20⋊6C4, C10.Q16, C5×C4.Q8, C4⋊D20, D10⋊2Q8, D5×C2×C8, C2×C40⋊C2, C8⋊8D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, D20 [×2], C22×D5, C8⋊8D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D5×SD16, SD16⋊3D5, C8⋊8D20
►On 160 points
Generators in S160
(1 125 117 91 30 151 79 59)(2 92 80 126 31 60 118 152)(3 127 119 93 32 153 61 41)(4 94 62 128 33 42 120 154)(5 129 101 95 34 155 63 43)(6 96 64 130 35 44 102 156)(7 131 103 97 36 157 65 45)(8 98 66 132 37 46 104 158)(9 133 105 99 38 159 67 47)(10 100 68 134 39 48 106 160)(11 135 107 81 40 141 69 49)(12 82 70 136 21 50 108 142)(13 137 109 83 22 143 71 51)(14 84 72 138 23 52 110 144)(15 139 111 85 24 145 73 53)(16 86 74 140 25 54 112 146)(17 121 113 87 26 147 75 55)(18 88 76 122 27 56 114 148)(19 123 115 89 28 149 77 57)(20 90 78 124 29 58 116 150)
(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 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(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)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 120)(71 119)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 140)(91 139)(92 138)(93 137)(94 136)(95 135)(96 134)(97 133)(98 132)(99 131)(100 130)
G:=sub<Sym(160)| (1,125,117,91,30,151,79,59)(2,92,80,126,31,60,118,152)(3,127,119,93,32,153,61,41)(4,94,62,128,33,42,120,154)(5,129,101,95,34,155,63,43)(6,96,64,130,35,44,102,156)(7,131,103,97,36,157,65,45)(8,98,66,132,37,46,104,158)(9,133,105,99,38,159,67,47)(10,100,68,134,39,48,106,160)(11,135,107,81,40,141,69,49)(12,82,70,136,21,50,108,142)(13,137,109,83,22,143,71,51)(14,84,72,138,23,52,110,144)(15,139,111,85,24,145,73,53)(16,86,74,140,25,54,112,146)(17,121,113,87,26,147,75,55)(18,88,76,122,27,56,114,148)(19,123,115,89,28,149,77,57)(20,90,78,124,29,58,116,150), (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)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(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)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,120)(71,119)(72,118)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,131)(100,130)>;
G:=Group( (1,125,117,91,30,151,79,59)(2,92,80,126,31,60,118,152)(3,127,119,93,32,153,61,41)(4,94,62,128,33,42,120,154)(5,129,101,95,34,155,63,43)(6,96,64,130,35,44,102,156)(7,131,103,97,36,157,65,45)(8,98,66,132,37,46,104,158)(9,133,105,99,38,159,67,47)(10,100,68,134,39,48,106,160)(11,135,107,81,40,141,69,49)(12,82,70,136,21,50,108,142)(13,137,109,83,22,143,71,51)(14,84,72,138,23,52,110,144)(15,139,111,85,24,145,73,53)(16,86,74,140,25,54,112,146)(17,121,113,87,26,147,75,55)(18,88,76,122,27,56,114,148)(19,123,115,89,28,149,77,57)(20,90,78,124,29,58,116,150), (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)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(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)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,120)(71,119)(72,118)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,131)(100,130) );
G=PermutationGroup([(1,125,117,91,30,151,79,59),(2,92,80,126,31,60,118,152),(3,127,119,93,32,153,61,41),(4,94,62,128,33,42,120,154),(5,129,101,95,34,155,63,43),(6,96,64,130,35,44,102,156),(7,131,103,97,36,157,65,45),(8,98,66,132,37,46,104,158),(9,133,105,99,38,159,67,47),(10,100,68,134,39,48,106,160),(11,135,107,81,40,141,69,49),(12,82,70,136,21,50,108,142),(13,137,109,83,22,143,71,51),(14,84,72,138,23,52,110,144),(15,139,111,85,24,145,73,53),(16,86,74,140,25,54,112,146),(17,121,113,87,26,147,75,55),(18,88,76,122,27,56,114,148),(19,123,115,89,28,149,77,57),(20,90,78,124,29,58,116,150)], [(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),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(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),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,120),(71,119),(72,118),(73,117),(74,116),(75,115),(76,114),(77,113),(78,112),(79,111),(80,110),(81,129),(82,128),(83,127),(84,126),(85,125),(86,124),(87,123),(88,122),(89,121),(90,140),(91,139),(92,138),(93,137),(94,136),(95,135),(96,134),(97,133),(98,132),(99,131),(100,130)])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 4 | 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 | 2 | 2 | 8 | 8 | 10 | 10 | 40 | 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 | 1 | 1 | 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 | C4○D8 | D20 | Q8⋊2D5 | D4×D5 | D5×SD16 | SD16⋊3D5 |
| kernel | C8⋊8D20 | D20⋊6C4 | C10.Q16 | C5×C4.Q8 | C4⋊D20 | D10⋊2Q8 | D5×C2×C8 | C2×C40⋊C2 | C40 | C2×Dic5 | C22×D5 | C4.Q8 | C20 | D10 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C8⋊8D20 ►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 | 30 |
| 0 | 0 | 0 | 0 | 26 | 30 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 5 | 0 | 0 |
| 0 | 0 | 16 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,30,30],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,25,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
C8⋊8D20 in GAP, Magma, Sage, TeX
C_8\rtimes_8D_{20} % in TeX
G:=Group("C8:8D20"); // GroupNames label
G:=SmallGroup(320,491);
// by ID
G=gap.SmallGroup(320,491);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=c*a*c=a^3,c*b*c=b^-1>;
// generators/relations