metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊8Q8, Dic10⋊12D4, C42.173D10, C10.362- (1+4), C5⋊4(D4×Q8), C4⋊1(Q8×D5), C4⋊Q8⋊11D5, C20⋊3(C2×Q8), C20⋊Q8⋊44C2, D10⋊7(C2×Q8), C4.74(D4×D5), C20.72(C2×D4), C4⋊C4.218D10, (C4×D20).26C2, D10⋊Q8⋊48C2, D10⋊3Q8⋊36C2, (C4×Dic10)⋊52C2, (C2×Q8).146D10, Dic5.54(C2×D4), D20⋊8C4.13C2, C10.47(C22×Q8), (C2×C10).271C24, (C4×C20).212C22, (C2×C20).104C23, C10.101(C22×D4), (C2×D20).280C22, C4⋊Dic5.385C22, (Q8×C10).138C22, C22.292(C23×D5), (C2×Dic5).142C23, (C4×Dic5).168C22, (C22×D5).242C23, D10⋊C4.152C22, C2.37(Q8.10D10), (C2×Dic10).195C22, C10.D4.166C22, (C2×Q8×D5)⋊13C2, C2.74(C2×D4×D5), C2.30(C2×Q8×D5), (C5×C4⋊Q8)⋊13C2, (C2×C4×D5).154C22, (C5×C4⋊C4).214C22, (C2×C4).218(C22×D5), SmallGroup(320,1399)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 966 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×13], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×18], D4 [×4], Q8 [×16], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8 [×13], Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×5], D10 [×4], D10 [×4], C2×C10, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8, C4⋊Q8 [×2], C22×Q8 [×2], Dic10 [×4], Dic10 [×8], C4×D5 [×12], D20 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×2], D4×Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×4], C2×C4×D5 [×6], C2×D20, Q8×D5 [×8], Q8×C10 [×2], C4×Dic10, C4×D20, C20⋊Q8 [×2], D20⋊8C4 [×2], D10⋊Q8 [×4], D10⋊3Q8 [×2], C5×C4⋊Q8, C2×Q8×D5 [×2], D20⋊8Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- (1+4), C22×D5 [×7], D4×Q8, D4×D5 [×2], Q8×D5 [×2], C23×D5, C2×D4×D5, C2×Q8×D5, Q8.10D10, D20⋊8Q8
Generators and relations
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >
►On 160 points
Generators in S160
(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 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 91)(82 90)(83 89)(84 88)(85 87)(92 100)(93 99)(94 98)(95 97)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(121 131)(122 130)(123 129)(124 128)(125 127)(132 140)(133 139)(134 138)(135 137)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)
(1 89 129 154)(2 90 130 155)(3 91 131 156)(4 92 132 157)(5 93 133 158)(6 94 134 159)(7 95 135 160)(8 96 136 141)(9 97 137 142)(10 98 138 143)(11 99 139 144)(12 100 140 145)(13 81 121 146)(14 82 122 147)(15 83 123 148)(16 84 124 149)(17 85 125 150)(18 86 126 151)(19 87 127 152)(20 88 128 153)(21 80 50 106)(22 61 51 107)(23 62 52 108)(24 63 53 109)(25 64 54 110)(26 65 55 111)(27 66 56 112)(28 67 57 113)(29 68 58 114)(30 69 59 115)(31 70 60 116)(32 71 41 117)(33 72 42 118)(34 73 43 119)(35 74 44 120)(36 75 45 101)(37 76 46 102)(38 77 47 103)(39 78 48 104)(40 79 49 105)
(1 23 129 52)(2 34 130 43)(3 25 131 54)(4 36 132 45)(5 27 133 56)(6 38 134 47)(7 29 135 58)(8 40 136 49)(9 31 137 60)(10 22 138 51)(11 33 139 42)(12 24 140 53)(13 35 121 44)(14 26 122 55)(15 37 123 46)(16 28 124 57)(17 39 125 48)(18 30 126 59)(19 21 127 50)(20 32 128 41)(61 98 107 143)(62 89 108 154)(63 100 109 145)(64 91 110 156)(65 82 111 147)(66 93 112 158)(67 84 113 149)(68 95 114 160)(69 86 115 151)(70 97 116 142)(71 88 117 153)(72 99 118 144)(73 90 119 155)(74 81 120 146)(75 92 101 157)(76 83 102 148)(77 94 103 159)(78 85 104 150)(79 96 105 141)(80 87 106 152)
G:=sub<Sym(160)| (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)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,131)(122,130)(123,129)(124,128)(125,127)(132,140)(133,139)(134,138)(135,137)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152), (1,89,129,154)(2,90,130,155)(3,91,131,156)(4,92,132,157)(5,93,133,158)(6,94,134,159)(7,95,135,160)(8,96,136,141)(9,97,137,142)(10,98,138,143)(11,99,139,144)(12,100,140,145)(13,81,121,146)(14,82,122,147)(15,83,123,148)(16,84,124,149)(17,85,125,150)(18,86,126,151)(19,87,127,152)(20,88,128,153)(21,80,50,106)(22,61,51,107)(23,62,52,108)(24,63,53,109)(25,64,54,110)(26,65,55,111)(27,66,56,112)(28,67,57,113)(29,68,58,114)(30,69,59,115)(31,70,60,116)(32,71,41,117)(33,72,42,118)(34,73,43,119)(35,74,44,120)(36,75,45,101)(37,76,46,102)(38,77,47,103)(39,78,48,104)(40,79,49,105), (1,23,129,52)(2,34,130,43)(3,25,131,54)(4,36,132,45)(5,27,133,56)(6,38,134,47)(7,29,135,58)(8,40,136,49)(9,31,137,60)(10,22,138,51)(11,33,139,42)(12,24,140,53)(13,35,121,44)(14,26,122,55)(15,37,123,46)(16,28,124,57)(17,39,125,48)(18,30,126,59)(19,21,127,50)(20,32,128,41)(61,98,107,143)(62,89,108,154)(63,100,109,145)(64,91,110,156)(65,82,111,147)(66,93,112,158)(67,84,113,149)(68,95,114,160)(69,86,115,151)(70,97,116,142)(71,88,117,153)(72,99,118,144)(73,90,119,155)(74,81,120,146)(75,92,101,157)(76,83,102,148)(77,94,103,159)(78,85,104,150)(79,96,105,141)(80,87,106,152)>;
G:=Group( (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)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,112)(109,111)(121,131)(122,130)(123,129)(124,128)(125,127)(132,140)(133,139)(134,138)(135,137)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152), (1,89,129,154)(2,90,130,155)(3,91,131,156)(4,92,132,157)(5,93,133,158)(6,94,134,159)(7,95,135,160)(8,96,136,141)(9,97,137,142)(10,98,138,143)(11,99,139,144)(12,100,140,145)(13,81,121,146)(14,82,122,147)(15,83,123,148)(16,84,124,149)(17,85,125,150)(18,86,126,151)(19,87,127,152)(20,88,128,153)(21,80,50,106)(22,61,51,107)(23,62,52,108)(24,63,53,109)(25,64,54,110)(26,65,55,111)(27,66,56,112)(28,67,57,113)(29,68,58,114)(30,69,59,115)(31,70,60,116)(32,71,41,117)(33,72,42,118)(34,73,43,119)(35,74,44,120)(36,75,45,101)(37,76,46,102)(38,77,47,103)(39,78,48,104)(40,79,49,105), (1,23,129,52)(2,34,130,43)(3,25,131,54)(4,36,132,45)(5,27,133,56)(6,38,134,47)(7,29,135,58)(8,40,136,49)(9,31,137,60)(10,22,138,51)(11,33,139,42)(12,24,140,53)(13,35,121,44)(14,26,122,55)(15,37,123,46)(16,28,124,57)(17,39,125,48)(18,30,126,59)(19,21,127,50)(20,32,128,41)(61,98,107,143)(62,89,108,154)(63,100,109,145)(64,91,110,156)(65,82,111,147)(66,93,112,158)(67,84,113,149)(68,95,114,160)(69,86,115,151)(70,97,116,142)(71,88,117,153)(72,99,118,144)(73,90,119,155)(74,81,120,146)(75,92,101,157)(76,83,102,148)(77,94,103,159)(78,85,104,150)(79,96,105,141)(80,87,106,152) );
G=PermutationGroup([(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),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,91),(82,90),(83,89),(84,88),(85,87),(92,100),(93,99),(94,98),(95,97),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,112),(109,111),(121,131),(122,130),(123,129),(124,128),(125,127),(132,140),(133,139),(134,138),(135,137),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152)], [(1,89,129,154),(2,90,130,155),(3,91,131,156),(4,92,132,157),(5,93,133,158),(6,94,134,159),(7,95,135,160),(8,96,136,141),(9,97,137,142),(10,98,138,143),(11,99,139,144),(12,100,140,145),(13,81,121,146),(14,82,122,147),(15,83,123,148),(16,84,124,149),(17,85,125,150),(18,86,126,151),(19,87,127,152),(20,88,128,153),(21,80,50,106),(22,61,51,107),(23,62,52,108),(24,63,53,109),(25,64,54,110),(26,65,55,111),(27,66,56,112),(28,67,57,113),(29,68,58,114),(30,69,59,115),(31,70,60,116),(32,71,41,117),(33,72,42,118),(34,73,43,119),(35,74,44,120),(36,75,45,101),(37,76,46,102),(38,77,47,103),(39,78,48,104),(40,79,49,105)], [(1,23,129,52),(2,34,130,43),(3,25,131,54),(4,36,132,45),(5,27,133,56),(6,38,134,47),(7,29,135,58),(8,40,136,49),(9,31,137,60),(10,22,138,51),(11,33,139,42),(12,24,140,53),(13,35,121,44),(14,26,122,55),(15,37,123,46),(16,28,124,57),(17,39,125,48),(18,30,126,59),(19,21,127,50),(20,32,128,41),(61,98,107,143),(62,89,108,154),(63,100,109,145),(64,91,110,156),(65,82,111,147),(66,93,112,158),(67,84,113,149),(68,95,114,160),(69,86,115,151),(70,97,116,142),(71,88,117,153),(72,99,118,144),(73,90,119,155),(74,81,120,146),(75,92,101,157),(76,83,102,148),(77,94,103,159),(78,85,104,150),(79,96,105,141),(80,87,106,152)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 8 | 4 | 0 | 0 | 0 | 0 |
| 35 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 37 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 33 | 37 | 0 | 0 | 0 | 0 |
| 26 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 1 |
| 0 | 0 | 0 | 0 | 1 | 11 |
G:=sub<GL(6,GF(41))| [8,35,0,0,0,0,4,33,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,37,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[33,26,0,0,0,0,37,8,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,1,0,0,0,0,1,11] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8×D5 | Q8.10D10 |
| kernel | D20⋊8Q8 | C4×Dic10 | C4×D20 | C20⋊Q8 | D20⋊8C4 | D10⋊Q8 | D10⋊3Q8 | C5×C4⋊Q8 | C2×Q8×D5 | Dic10 | D20 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_8Q_8 % in TeX
G:=Group("D20:8Q8"); // GroupNames label
G:=SmallGroup(320,1399);
// by ID
G=gap.SmallGroup(320,1399);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀