metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.164D10, C10.1022- 1+4, C10.1412+ 1+4, C20⋊Q8⋊41C2, C4⋊C4.119D10, C4.D20⋊9C2, C42⋊2C2⋊7D5, D10⋊2Q8⋊42C2, D10⋊Q8⋊45C2, (C4×Dic10)⋊16C2, D10⋊D4.5C2, (C4×C20).36C22, C22⋊C4.82D10, Dic5⋊4D4⋊38C2, (C2×C10).254C24, (C2×C20).605C23, (C2×D20).39C22, D10.13D4⋊43C2, C2.66(D4⋊8D10), C23.60(C22×D5), Dic5.22(C4○D4), Dic5.5D4⋊47C2, C22.D20⋊31C2, C4⋊Dic5.319C22, (C22×C10).68C23, C22.275(C23×D5), Dic5.14D4⋊47C2, C23.D5.69C22, D10⋊C4.47C22, (C4×Dic5).239C22, (C2×Dic5).277C23, (C22×D5).113C23, C2.66(D4.10D10), C5⋊10(C22.36C24), (C2×Dic10).264C22, C10.D4.127C22, (C22×Dic5).154C22, C4⋊C4⋊7D5⋊42C2, C4⋊C4⋊D5⋊44C2, C2.101(D5×C4○D4), (C5×C42⋊2C2)⋊9C2, C10.212(C2×C4○D4), (C2×C4×D5).145C22, (C5×C4⋊C4).206C22, (C2×C4).210(C22×D5), (C2×C5⋊D4).74C22, (C5×C22⋊C4).79C22, SmallGroup(320,1382)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.164D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 798 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C22.36C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C4×Dic10, C4.D20, Dic5.14D4, Dic5⋊4D4, D10⋊D4, Dic5.5D4, C22.D20, C20⋊Q8, C4⋊C4⋊7D5, D10.13D4, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.164D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, C23×D5, D5×C4○D4, D4⋊8D10, D4.10D10, C42.164D10
►On 160 points
Generators in S160
(1 88 73 50)(2 99 74 41)(3 90 75 52)(4 81 76 43)(5 92 77 54)(6 83 78 45)(7 94 79 56)(8 85 80 47)(9 96 61 58)(10 87 62 49)(11 98 63 60)(12 89 64 51)(13 100 65 42)(14 91 66 53)(15 82 67 44)(16 93 68 55)(17 84 69 46)(18 95 70 57)(19 86 71 48)(20 97 72 59)(21 135 144 116)(22 126 145 107)(23 137 146 118)(24 128 147 109)(25 139 148 120)(26 130 149 111)(27 121 150 102)(28 132 151 113)(29 123 152 104)(30 134 153 115)(31 125 154 106)(32 136 155 117)(33 127 156 108)(34 138 157 119)(35 129 158 110)(36 140 159 101)(37 131 160 112)(38 122 141 103)(39 133 142 114)(40 124 143 105)
(1 135 11 125)(2 107 12 117)(3 137 13 127)(4 109 14 119)(5 139 15 129)(6 111 16 101)(7 121 17 131)(8 113 18 103)(9 123 19 133)(10 115 20 105)(21 60 31 50)(22 89 32 99)(23 42 33 52)(24 91 34 81)(25 44 35 54)(26 93 36 83)(27 46 37 56)(28 95 38 85)(29 48 39 58)(30 97 40 87)(41 145 51 155)(43 147 53 157)(45 149 55 159)(47 151 57 141)(49 153 59 143)(61 104 71 114)(62 134 72 124)(63 106 73 116)(64 136 74 126)(65 108 75 118)(66 138 76 128)(67 110 77 120)(68 140 78 130)(69 112 79 102)(70 122 80 132)(82 158 92 148)(84 160 94 150)(86 142 96 152)(88 144 98 154)(90 146 100 156)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 153 31 143)(22 142 32 152)(23 151 33 141)(24 160 34 150)(25 149 35 159)(26 158 36 148)(27 147 37 157)(28 156 38 146)(29 145 39 155)(30 154 40 144)(41 86 51 96)(42 95 52 85)(43 84 53 94)(44 93 54 83)(45 82 55 92)(46 91 56 81)(47 100 57 90)(48 89 58 99)(49 98 59 88)(50 87 60 97)(61 74 71 64)(62 63 72 73)(65 70 75 80)(66 79 76 69)(67 68 77 78)(101 120 111 110)(102 109 112 119)(103 118 113 108)(104 107 114 117)(105 116 115 106)(121 128 131 138)(122 137 132 127)(123 126 133 136)(124 135 134 125)(129 140 139 130)
G:=sub<Sym(160)| (1,88,73,50)(2,99,74,41)(3,90,75,52)(4,81,76,43)(5,92,77,54)(6,83,78,45)(7,94,79,56)(8,85,80,47)(9,96,61,58)(10,87,62,49)(11,98,63,60)(12,89,64,51)(13,100,65,42)(14,91,66,53)(15,82,67,44)(16,93,68,55)(17,84,69,46)(18,95,70,57)(19,86,71,48)(20,97,72,59)(21,135,144,116)(22,126,145,107)(23,137,146,118)(24,128,147,109)(25,139,148,120)(26,130,149,111)(27,121,150,102)(28,132,151,113)(29,123,152,104)(30,134,153,115)(31,125,154,106)(32,136,155,117)(33,127,156,108)(34,138,157,119)(35,129,158,110)(36,140,159,101)(37,131,160,112)(38,122,141,103)(39,133,142,114)(40,124,143,105), (1,135,11,125)(2,107,12,117)(3,137,13,127)(4,109,14,119)(5,139,15,129)(6,111,16,101)(7,121,17,131)(8,113,18,103)(9,123,19,133)(10,115,20,105)(21,60,31,50)(22,89,32,99)(23,42,33,52)(24,91,34,81)(25,44,35,54)(26,93,36,83)(27,46,37,56)(28,95,38,85)(29,48,39,58)(30,97,40,87)(41,145,51,155)(43,147,53,157)(45,149,55,159)(47,151,57,141)(49,153,59,143)(61,104,71,114)(62,134,72,124)(63,106,73,116)(64,136,74,126)(65,108,75,118)(66,138,76,128)(67,110,77,120)(68,140,78,130)(69,112,79,102)(70,122,80,132)(82,158,92,148)(84,160,94,150)(86,142,96,152)(88,144,98,154)(90,146,100,156), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,153,31,143)(22,142,32,152)(23,151,33,141)(24,160,34,150)(25,149,35,159)(26,158,36,148)(27,147,37,157)(28,156,38,146)(29,145,39,155)(30,154,40,144)(41,86,51,96)(42,95,52,85)(43,84,53,94)(44,93,54,83)(45,82,55,92)(46,91,56,81)(47,100,57,90)(48,89,58,99)(49,98,59,88)(50,87,60,97)(61,74,71,64)(62,63,72,73)(65,70,75,80)(66,79,76,69)(67,68,77,78)(101,120,111,110)(102,109,112,119)(103,118,113,108)(104,107,114,117)(105,116,115,106)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130)>;
G:=Group( (1,88,73,50)(2,99,74,41)(3,90,75,52)(4,81,76,43)(5,92,77,54)(6,83,78,45)(7,94,79,56)(8,85,80,47)(9,96,61,58)(10,87,62,49)(11,98,63,60)(12,89,64,51)(13,100,65,42)(14,91,66,53)(15,82,67,44)(16,93,68,55)(17,84,69,46)(18,95,70,57)(19,86,71,48)(20,97,72,59)(21,135,144,116)(22,126,145,107)(23,137,146,118)(24,128,147,109)(25,139,148,120)(26,130,149,111)(27,121,150,102)(28,132,151,113)(29,123,152,104)(30,134,153,115)(31,125,154,106)(32,136,155,117)(33,127,156,108)(34,138,157,119)(35,129,158,110)(36,140,159,101)(37,131,160,112)(38,122,141,103)(39,133,142,114)(40,124,143,105), (1,135,11,125)(2,107,12,117)(3,137,13,127)(4,109,14,119)(5,139,15,129)(6,111,16,101)(7,121,17,131)(8,113,18,103)(9,123,19,133)(10,115,20,105)(21,60,31,50)(22,89,32,99)(23,42,33,52)(24,91,34,81)(25,44,35,54)(26,93,36,83)(27,46,37,56)(28,95,38,85)(29,48,39,58)(30,97,40,87)(41,145,51,155)(43,147,53,157)(45,149,55,159)(47,151,57,141)(49,153,59,143)(61,104,71,114)(62,134,72,124)(63,106,73,116)(64,136,74,126)(65,108,75,118)(66,138,76,128)(67,110,77,120)(68,140,78,130)(69,112,79,102)(70,122,80,132)(82,158,92,148)(84,160,94,150)(86,142,96,152)(88,144,98,154)(90,146,100,156), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,153,31,143)(22,142,32,152)(23,151,33,141)(24,160,34,150)(25,149,35,159)(26,158,36,148)(27,147,37,157)(28,156,38,146)(29,145,39,155)(30,154,40,144)(41,86,51,96)(42,95,52,85)(43,84,53,94)(44,93,54,83)(45,82,55,92)(46,91,56,81)(47,100,57,90)(48,89,58,99)(49,98,59,88)(50,87,60,97)(61,74,71,64)(62,63,72,73)(65,70,75,80)(66,79,76,69)(67,68,77,78)(101,120,111,110)(102,109,112,119)(103,118,113,108)(104,107,114,117)(105,116,115,106)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130) );
G=PermutationGroup([[(1,88,73,50),(2,99,74,41),(3,90,75,52),(4,81,76,43),(5,92,77,54),(6,83,78,45),(7,94,79,56),(8,85,80,47),(9,96,61,58),(10,87,62,49),(11,98,63,60),(12,89,64,51),(13,100,65,42),(14,91,66,53),(15,82,67,44),(16,93,68,55),(17,84,69,46),(18,95,70,57),(19,86,71,48),(20,97,72,59),(21,135,144,116),(22,126,145,107),(23,137,146,118),(24,128,147,109),(25,139,148,120),(26,130,149,111),(27,121,150,102),(28,132,151,113),(29,123,152,104),(30,134,153,115),(31,125,154,106),(32,136,155,117),(33,127,156,108),(34,138,157,119),(35,129,158,110),(36,140,159,101),(37,131,160,112),(38,122,141,103),(39,133,142,114),(40,124,143,105)], [(1,135,11,125),(2,107,12,117),(3,137,13,127),(4,109,14,119),(5,139,15,129),(6,111,16,101),(7,121,17,131),(8,113,18,103),(9,123,19,133),(10,115,20,105),(21,60,31,50),(22,89,32,99),(23,42,33,52),(24,91,34,81),(25,44,35,54),(26,93,36,83),(27,46,37,56),(28,95,38,85),(29,48,39,58),(30,97,40,87),(41,145,51,155),(43,147,53,157),(45,149,55,159),(47,151,57,141),(49,153,59,143),(61,104,71,114),(62,134,72,124),(63,106,73,116),(64,136,74,126),(65,108,75,118),(66,138,76,128),(67,110,77,120),(68,140,78,130),(69,112,79,102),(70,122,80,132),(82,158,92,148),(84,160,94,150),(86,142,96,152),(88,144,98,154),(90,146,100,156)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,153,31,143),(22,142,32,152),(23,151,33,141),(24,160,34,150),(25,149,35,159),(26,158,36,148),(27,147,37,157),(28,156,38,146),(29,145,39,155),(30,154,40,144),(41,86,51,96),(42,95,52,85),(43,84,53,94),(44,93,54,83),(45,82,55,92),(46,91,56,81),(47,100,57,90),(48,89,58,99),(49,98,59,88),(50,87,60,97),(61,74,71,64),(62,63,72,73),(65,70,75,80),(66,79,76,69),(67,68,77,78),(101,120,111,110),(102,109,112,119),(103,118,113,108),(104,107,114,117),(105,116,115,106),(121,128,131,138),(122,137,132,127),(123,126,133,136),(124,135,134,125),(129,140,139,130)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D5×C4○D4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.164D10 | C4×Dic10 | C4.D20 | Dic5.14D4 | Dic5⋊4D4 | D10⋊D4 | Dic5.5D4 | C22.D20 | C20⋊Q8 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 6 | 6 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.164D10 ►in GL6(𝔽41)
| 32 | 39 | 0 | 0 | 0 | 0 |
| 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 0 | 28 | 39 |
| 40 | 18 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 1 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 39 | 39 |
| 0 | 0 | 9 | 19 | 2 | 27 |
| 0 | 0 | 2 | 2 | 9 | 9 |
| 0 | 0 | 39 | 14 | 32 | 22 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 39 | 39 |
| 0 | 0 | 19 | 9 | 27 | 2 |
| 0 | 0 | 2 | 2 | 9 | 9 |
| 0 | 0 | 14 | 39 | 22 | 32 |
G:=sub<GL(6,GF(41))| [32,40,0,0,0,0,39,9,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,9,0,0,0,0,18,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[9,1,0,0,0,0,0,32,0,0,0,0,0,0,32,9,2,39,0,0,32,19,2,14,0,0,39,2,9,32,0,0,39,27,9,22],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,19,2,14,0,0,32,9,2,39,0,0,39,27,9,22,0,0,39,2,9,32] >;
C42.164D10 in GAP, Magma, Sage, TeX
C_4^2._{164}D_{10} % in TeX
G:=Group("C4^2.164D10"); // GroupNames label
G:=SmallGroup(320,1382);
// by ID
G=gap.SmallGroup(320,1382);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations