metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.140D10, C10.892- 1+4, C10.722+ 1+4, (C2×Q8).83D10, C4.4D4.9D5, (C2×D4).109D10, (C2×C20).78C23, C22⋊C4.34D10, C20.6Q8⋊28C2, Dic5⋊Q8⋊23C2, (C4×C20).221C22, (C2×C10).216C24, C4⋊Dic5.50C22, C2.74(D4⋊6D10), C23.38(C22×D5), (D4×C10).209C22, C23.D10⋊38C2, (C22×C10).46C23, (Q8×C10).125C22, C22.237(C23×D5), Dic5.14D4⋊39C2, C23.D5.53C22, C5⋊3(C22.57C24), (C2×Dic5).111C23, (C4×Dic5).140C22, C10.D4.83C22, C23.18D10.6C2, C2.50(D4.10D10), (C2×Dic10).182C22, (C22×Dic5).141C22, (C5×C4.4D4).7C2, (C2×C4).192(C22×D5), (C5×C22⋊C4).63C22, SmallGroup(320,1344)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.140D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 614 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C2×C10, C2×C10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.57C24, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C22×Dic5, D4×C10, Q8×C10, C20.6Q8, Dic5.14D4, C23.D10, C23.18D10, Dic5⋊Q8, C5×C4.4D4, C42.140D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.57C24, C23×D5, D4⋊6D10, D4.10D10, C42.140D10
►On 160 points
Generators in S160
(1 100 8 127)(2 96 9 123)(3 92 10 129)(4 98 6 125)(5 94 7 121)(11 126 36 99)(12 122 37 95)(13 128 38 91)(14 124 39 97)(15 130 40 93)(16 61 31 42)(17 67 32 48)(18 63 33 44)(19 69 34 50)(20 65 35 46)(21 47 26 66)(22 43 27 62)(23 49 28 68)(24 45 29 64)(25 41 30 70)(51 103 56 158)(52 154 57 109)(53 105 58 160)(54 156 59 101)(55 107 60 152)(71 106 76 151)(72 157 77 102)(73 108 78 153)(74 159 79 104)(75 110 80 155)(81 144 86 116)(82 112 87 150)(83 146 88 118)(84 114 89 142)(85 148 90 120)(111 137 149 132)(113 139 141 134)(115 131 143 136)(117 133 145 138)(119 135 147 140)
(1 113 12 118)(2 119 13 114)(3 115 14 120)(4 111 15 116)(5 117 11 112)(6 149 40 144)(7 145 36 150)(8 141 37 146)(9 147 38 142)(10 143 39 148)(16 104 21 109)(17 110 22 105)(18 106 23 101)(19 102 24 107)(20 108 25 103)(26 154 31 159)(27 160 32 155)(28 156 33 151)(29 152 34 157)(30 158 35 153)(41 56 65 78)(42 79 66 57)(43 58 67 80)(44 71 68 59)(45 60 69 72)(46 73 70 51)(47 52 61 74)(48 75 62 53)(49 54 63 76)(50 77 64 55)(81 98 137 130)(82 121 138 99)(83 100 139 122)(84 123 140 91)(85 92 131 124)(86 125 132 93)(87 94 133 126)(88 127 134 95)(89 96 135 128)(90 129 136 97)
(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 28 12 33)(2 27 13 32)(3 26 14 31)(4 30 15 35)(5 29 11 34)(6 25 40 20)(7 24 36 19)(8 23 37 18)(9 22 38 17)(10 21 39 16)(41 130 65 98)(42 129 66 97)(43 128 67 96)(44 127 68 95)(45 126 69 94)(46 125 70 93)(47 124 61 92)(48 123 62 91)(49 122 63 100)(50 121 64 99)(51 137 73 81)(52 136 74 90)(53 135 75 89)(54 134 76 88)(55 133 77 87)(56 132 78 86)(57 131 79 85)(58 140 80 84)(59 139 71 83)(60 138 72 82)(101 113 106 118)(102 112 107 117)(103 111 108 116)(104 120 109 115)(105 119 110 114)(141 151 146 156)(142 160 147 155)(143 159 148 154)(144 158 149 153)(145 157 150 152)
G:=sub<Sym(160)| (1,100,8,127)(2,96,9,123)(3,92,10,129)(4,98,6,125)(5,94,7,121)(11,126,36,99)(12,122,37,95)(13,128,38,91)(14,124,39,97)(15,130,40,93)(16,61,31,42)(17,67,32,48)(18,63,33,44)(19,69,34,50)(20,65,35,46)(21,47,26,66)(22,43,27,62)(23,49,28,68)(24,45,29,64)(25,41,30,70)(51,103,56,158)(52,154,57,109)(53,105,58,160)(54,156,59,101)(55,107,60,152)(71,106,76,151)(72,157,77,102)(73,108,78,153)(74,159,79,104)(75,110,80,155)(81,144,86,116)(82,112,87,150)(83,146,88,118)(84,114,89,142)(85,148,90,120)(111,137,149,132)(113,139,141,134)(115,131,143,136)(117,133,145,138)(119,135,147,140), (1,113,12,118)(2,119,13,114)(3,115,14,120)(4,111,15,116)(5,117,11,112)(6,149,40,144)(7,145,36,150)(8,141,37,146)(9,147,38,142)(10,143,39,148)(16,104,21,109)(17,110,22,105)(18,106,23,101)(19,102,24,107)(20,108,25,103)(26,154,31,159)(27,160,32,155)(28,156,33,151)(29,152,34,157)(30,158,35,153)(41,56,65,78)(42,79,66,57)(43,58,67,80)(44,71,68,59)(45,60,69,72)(46,73,70,51)(47,52,61,74)(48,75,62,53)(49,54,63,76)(50,77,64,55)(81,98,137,130)(82,121,138,99)(83,100,139,122)(84,123,140,91)(85,92,131,124)(86,125,132,93)(87,94,133,126)(88,127,134,95)(89,96,135,128)(90,129,136,97), (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), (1,28,12,33)(2,27,13,32)(3,26,14,31)(4,30,15,35)(5,29,11,34)(6,25,40,20)(7,24,36,19)(8,23,37,18)(9,22,38,17)(10,21,39,16)(41,130,65,98)(42,129,66,97)(43,128,67,96)(44,127,68,95)(45,126,69,94)(46,125,70,93)(47,124,61,92)(48,123,62,91)(49,122,63,100)(50,121,64,99)(51,137,73,81)(52,136,74,90)(53,135,75,89)(54,134,76,88)(55,133,77,87)(56,132,78,86)(57,131,79,85)(58,140,80,84)(59,139,71,83)(60,138,72,82)(101,113,106,118)(102,112,107,117)(103,111,108,116)(104,120,109,115)(105,119,110,114)(141,151,146,156)(142,160,147,155)(143,159,148,154)(144,158,149,153)(145,157,150,152)>;
G:=Group( (1,100,8,127)(2,96,9,123)(3,92,10,129)(4,98,6,125)(5,94,7,121)(11,126,36,99)(12,122,37,95)(13,128,38,91)(14,124,39,97)(15,130,40,93)(16,61,31,42)(17,67,32,48)(18,63,33,44)(19,69,34,50)(20,65,35,46)(21,47,26,66)(22,43,27,62)(23,49,28,68)(24,45,29,64)(25,41,30,70)(51,103,56,158)(52,154,57,109)(53,105,58,160)(54,156,59,101)(55,107,60,152)(71,106,76,151)(72,157,77,102)(73,108,78,153)(74,159,79,104)(75,110,80,155)(81,144,86,116)(82,112,87,150)(83,146,88,118)(84,114,89,142)(85,148,90,120)(111,137,149,132)(113,139,141,134)(115,131,143,136)(117,133,145,138)(119,135,147,140), (1,113,12,118)(2,119,13,114)(3,115,14,120)(4,111,15,116)(5,117,11,112)(6,149,40,144)(7,145,36,150)(8,141,37,146)(9,147,38,142)(10,143,39,148)(16,104,21,109)(17,110,22,105)(18,106,23,101)(19,102,24,107)(20,108,25,103)(26,154,31,159)(27,160,32,155)(28,156,33,151)(29,152,34,157)(30,158,35,153)(41,56,65,78)(42,79,66,57)(43,58,67,80)(44,71,68,59)(45,60,69,72)(46,73,70,51)(47,52,61,74)(48,75,62,53)(49,54,63,76)(50,77,64,55)(81,98,137,130)(82,121,138,99)(83,100,139,122)(84,123,140,91)(85,92,131,124)(86,125,132,93)(87,94,133,126)(88,127,134,95)(89,96,135,128)(90,129,136,97), (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), (1,28,12,33)(2,27,13,32)(3,26,14,31)(4,30,15,35)(5,29,11,34)(6,25,40,20)(7,24,36,19)(8,23,37,18)(9,22,38,17)(10,21,39,16)(41,130,65,98)(42,129,66,97)(43,128,67,96)(44,127,68,95)(45,126,69,94)(46,125,70,93)(47,124,61,92)(48,123,62,91)(49,122,63,100)(50,121,64,99)(51,137,73,81)(52,136,74,90)(53,135,75,89)(54,134,76,88)(55,133,77,87)(56,132,78,86)(57,131,79,85)(58,140,80,84)(59,139,71,83)(60,138,72,82)(101,113,106,118)(102,112,107,117)(103,111,108,116)(104,120,109,115)(105,119,110,114)(141,151,146,156)(142,160,147,155)(143,159,148,154)(144,158,149,153)(145,157,150,152) );
G=PermutationGroup([[(1,100,8,127),(2,96,9,123),(3,92,10,129),(4,98,6,125),(5,94,7,121),(11,126,36,99),(12,122,37,95),(13,128,38,91),(14,124,39,97),(15,130,40,93),(16,61,31,42),(17,67,32,48),(18,63,33,44),(19,69,34,50),(20,65,35,46),(21,47,26,66),(22,43,27,62),(23,49,28,68),(24,45,29,64),(25,41,30,70),(51,103,56,158),(52,154,57,109),(53,105,58,160),(54,156,59,101),(55,107,60,152),(71,106,76,151),(72,157,77,102),(73,108,78,153),(74,159,79,104),(75,110,80,155),(81,144,86,116),(82,112,87,150),(83,146,88,118),(84,114,89,142),(85,148,90,120),(111,137,149,132),(113,139,141,134),(115,131,143,136),(117,133,145,138),(119,135,147,140)], [(1,113,12,118),(2,119,13,114),(3,115,14,120),(4,111,15,116),(5,117,11,112),(6,149,40,144),(7,145,36,150),(8,141,37,146),(9,147,38,142),(10,143,39,148),(16,104,21,109),(17,110,22,105),(18,106,23,101),(19,102,24,107),(20,108,25,103),(26,154,31,159),(27,160,32,155),(28,156,33,151),(29,152,34,157),(30,158,35,153),(41,56,65,78),(42,79,66,57),(43,58,67,80),(44,71,68,59),(45,60,69,72),(46,73,70,51),(47,52,61,74),(48,75,62,53),(49,54,63,76),(50,77,64,55),(81,98,137,130),(82,121,138,99),(83,100,139,122),(84,123,140,91),(85,92,131,124),(86,125,132,93),(87,94,133,126),(88,127,134,95),(89,96,135,128),(90,129,136,97)], [(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)], [(1,28,12,33),(2,27,13,32),(3,26,14,31),(4,30,15,35),(5,29,11,34),(6,25,40,20),(7,24,36,19),(8,23,37,18),(9,22,38,17),(10,21,39,16),(41,130,65,98),(42,129,66,97),(43,128,67,96),(44,127,68,95),(45,126,69,94),(46,125,70,93),(47,124,61,92),(48,123,62,91),(49,122,63,100),(50,121,64,99),(51,137,73,81),(52,136,74,90),(53,135,75,89),(54,134,76,88),(55,133,77,87),(56,132,78,86),(57,131,79,85),(58,140,80,84),(59,139,71,83),(60,138,72,82),(101,113,106,118),(102,112,107,117),(103,111,108,116),(104,120,109,115),(105,119,110,114),(141,151,146,156),(142,160,147,155),(143,159,148,154),(144,158,149,153),(145,157,150,152)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 4F | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | D4.10D10 |
| kernel | C42.140D10 | C20.6Q8 | Dic5.14D4 | C23.D10 | C23.18D10 | Dic5⋊Q8 | C5×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 2 | 4 | 8 |
Matrix representation of C42.140D10 ►in GL8(𝔽41)
| 11 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 0 | 2 | 32 | 0 | 0 | 0 | 0 |
| 28 | 9 | 37 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 | 33 | 35 |
| 0 | 0 | 0 | 0 | 28 | 39 | 33 | 39 |
| 0 | 0 | 0 | 0 | 36 | 15 | 30 | 28 |
| 0 | 0 | 0 | 0 | 20 | 21 | 22 | 11 |
| 0 | 0 | 34 | 1 | 0 | 0 | 0 | 0 |
| 40 | 40 | 39 | 40 | 0 | 0 | 0 | 0 |
| 23 | 23 | 1 | 0 | 0 | 0 | 0 | 0 |
| 37 | 38 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 38 | 38 |
| 0 | 0 | 0 | 0 | 0 | 1 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 40 | 0 |
| 0 | 0 | 0 | 0 | 28 | 28 | 0 | 40 |
| 40 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 6 | 35 | 34 | 0 | 0 | 0 | 0 |
| 16 | 8 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 28 | 7 | 6 |
| 0 | 0 | 0 | 0 | 0 | 9 | 34 | 0 |
| 24 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 0 | 20 | 17 | 0 | 0 | 0 | 0 |
| 38 | 1 | 15 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 35 | 19 | 28 |
| 0 | 0 | 0 | 0 | 40 | 6 | 31 | 22 |
G:=sub<GL(8,GF(41))| [11,9,32,28,0,0,0,0,32,30,0,9,0,0,0,0,0,0,2,37,0,0,0,0,0,0,32,39,0,0,0,0,0,0,0,0,2,28,36,20,0,0,0,0,13,39,15,21,0,0,0,0,33,33,30,22,0,0,0,0,35,39,28,11],[0,40,23,37,0,0,0,0,0,40,23,38,0,0,0,0,34,39,1,7,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,13,28,0,0,0,0,38,3,40,0,0,0,0,0,38,0,0,40],[40,34,13,16,0,0,0,0,7,7,6,8,0,0,0,0,0,0,35,6,0,0,0,0,0,0,34,0,0,0,0,0,0,0,0,0,35,6,37,0,0,0,0,0,35,40,28,9,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0],[24,3,17,38,0,0,0,0,40,17,0,1,0,0,0,0,0,0,20,15,0,0,0,0,0,0,17,21,0,0,0,0,0,0,0,0,32,28,36,40,0,0,0,0,0,9,35,6,0,0,0,0,0,0,19,31,0,0,0,0,0,0,28,22] >;
C42.140D10 in GAP, Magma, Sage, TeX
C_4^2._{140}D_{10} % in TeX
G:=Group("C4^2.140D10"); // GroupNames label
G:=SmallGroup(320,1344);
// by ID
G=gap.SmallGroup(320,1344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,219,184,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations