metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.241D10, C4⋊Q8⋊20D5, (C4×D5)⋊5Q8, C4.40(Q8×D5), D10.5(C2×Q8), C20.54(C2×Q8), C4⋊C4.219D10, C20⋊2Q8⋊36C2, (Q8×Dic5)⋊22C2, (C2×Q8).147D10, C4.Dic10⋊42C2, Dic5.34(C2×Q8), (D5×C42).10C2, Dic5⋊3Q8⋊42C2, C20.136(C4○D4), C4.41(D4⋊2D5), C10.48(C22×Q8), (C2×C20).105C23, (C4×C20).213C22, (C2×C10).272C24, D10⋊3Q8.12C2, D10⋊2Q8.14C2, C4.22(Q8⋊2D5), C4⋊Dic5.251C22, (Q8×C10).139C22, C22.293(C23×D5), D10⋊C4.51C22, C5⋊6(C23.37C23), (C2×Dic5).143C23, (C4×Dic5).169C22, C10.D4.61C22, (C22×D5).243C23, (C2×Dic10).196C22, C2.31(C2×Q8×D5), (C5×C4⋊Q8)⋊14C2, C4⋊C4⋊7D5.14C2, C10.100(C2×C4○D4), C2.64(C2×D4⋊2D5), C2.29(C2×Q8⋊2D5), (C2×C4×D5).322C22, (C5×C4⋊C4).215C22, (C2×C4).600(C22×D5), SmallGroup(320,1400)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.241D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c9 >
Subgroups: 654 in 222 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.37C23, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Q8×C10, C20⋊2Q8, D5×C42, Dic5⋊3Q8, C4.Dic10, C4⋊C4⋊7D5, D10⋊2Q8, Q8×Dic5, D10⋊3Q8, C5×C4⋊Q8, C42.241D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, C22×D5, C23.37C23, D4⋊2D5, Q8×D5, Q8⋊2D5, C23×D5, C2×D4⋊2D5, C2×Q8×D5, C2×Q8⋊2D5, C42.241D10
►On 160 points
Generators in S160
(1 126 32 93)(2 94 33 127)(3 128 34 95)(4 96 35 129)(5 130 36 97)(6 98 37 131)(7 132 38 99)(8 100 39 133)(9 134 40 81)(10 82 21 135)(11 136 22 83)(12 84 23 137)(13 138 24 85)(14 86 25 139)(15 140 26 87)(16 88 27 121)(17 122 28 89)(18 90 29 123)(19 124 30 91)(20 92 31 125)(41 105 69 160)(42 141 70 106)(43 107 71 142)(44 143 72 108)(45 109 73 144)(46 145 74 110)(47 111 75 146)(48 147 76 112)(49 113 77 148)(50 149 78 114)(51 115 79 150)(52 151 80 116)(53 117 61 152)(54 153 62 118)(55 119 63 154)(56 155 64 120)(57 101 65 156)(58 157 66 102)(59 103 67 158)(60 159 68 104)
(1 50 11 60)(2 41 12 51)(3 52 13 42)(4 43 14 53)(5 54 15 44)(6 45 16 55)(7 56 17 46)(8 47 18 57)(9 58 19 48)(10 49 20 59)(21 77 31 67)(22 68 32 78)(23 79 33 69)(24 70 34 80)(25 61 35 71)(26 72 36 62)(27 63 37 73)(28 74 38 64)(29 65 39 75)(30 76 40 66)(81 102 91 112)(82 113 92 103)(83 104 93 114)(84 115 94 105)(85 106 95 116)(86 117 96 107)(87 108 97 118)(88 119 98 109)(89 110 99 120)(90 101 100 111)(121 154 131 144)(122 145 132 155)(123 156 133 146)(124 147 134 157)(125 158 135 148)(126 149 136 159)(127 160 137 150)(128 151 138 141)(129 142 139 152)(130 153 140 143)
(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 22 31)(2 30 23 9)(3 8 24 29)(4 28 25 7)(5 6 26 27)(11 20 32 21)(12 40 33 19)(13 18 34 39)(14 38 35 17)(15 16 36 37)(41 66 79 48)(42 47 80 65)(43 64 61 46)(44 45 62 63)(49 78 67 60)(50 59 68 77)(51 76 69 58)(52 57 70 75)(53 74 71 56)(54 55 72 73)(81 94 124 137)(82 136 125 93)(83 92 126 135)(84 134 127 91)(85 90 128 133)(86 132 129 89)(87 88 130 131)(95 100 138 123)(96 122 139 99)(97 98 140 121)(101 141 146 116)(102 115 147 160)(103 159 148 114)(104 113 149 158)(105 157 150 112)(106 111 151 156)(107 155 152 110)(108 109 153 154)(117 145 142 120)(118 119 143 144)
G:=sub<Sym(160)| (1,126,32,93)(2,94,33,127)(3,128,34,95)(4,96,35,129)(5,130,36,97)(6,98,37,131)(7,132,38,99)(8,100,39,133)(9,134,40,81)(10,82,21,135)(11,136,22,83)(12,84,23,137)(13,138,24,85)(14,86,25,139)(15,140,26,87)(16,88,27,121)(17,122,28,89)(18,90,29,123)(19,124,30,91)(20,92,31,125)(41,105,69,160)(42,141,70,106)(43,107,71,142)(44,143,72,108)(45,109,73,144)(46,145,74,110)(47,111,75,146)(48,147,76,112)(49,113,77,148)(50,149,78,114)(51,115,79,150)(52,151,80,116)(53,117,61,152)(54,153,62,118)(55,119,63,154)(56,155,64,120)(57,101,65,156)(58,157,66,102)(59,103,67,158)(60,159,68,104), (1,50,11,60)(2,41,12,51)(3,52,13,42)(4,43,14,53)(5,54,15,44)(6,45,16,55)(7,56,17,46)(8,47,18,57)(9,58,19,48)(10,49,20,59)(21,77,31,67)(22,68,32,78)(23,79,33,69)(24,70,34,80)(25,61,35,71)(26,72,36,62)(27,63,37,73)(28,74,38,64)(29,65,39,75)(30,76,40,66)(81,102,91,112)(82,113,92,103)(83,104,93,114)(84,115,94,105)(85,106,95,116)(86,117,96,107)(87,108,97,118)(88,119,98,109)(89,110,99,120)(90,101,100,111)(121,154,131,144)(122,145,132,155)(123,156,133,146)(124,147,134,157)(125,158,135,148)(126,149,136,159)(127,160,137,150)(128,151,138,141)(129,142,139,152)(130,153,140,143), (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), (1,10,22,31)(2,30,23,9)(3,8,24,29)(4,28,25,7)(5,6,26,27)(11,20,32,21)(12,40,33,19)(13,18,34,39)(14,38,35,17)(15,16,36,37)(41,66,79,48)(42,47,80,65)(43,64,61,46)(44,45,62,63)(49,78,67,60)(50,59,68,77)(51,76,69,58)(52,57,70,75)(53,74,71,56)(54,55,72,73)(81,94,124,137)(82,136,125,93)(83,92,126,135)(84,134,127,91)(85,90,128,133)(86,132,129,89)(87,88,130,131)(95,100,138,123)(96,122,139,99)(97,98,140,121)(101,141,146,116)(102,115,147,160)(103,159,148,114)(104,113,149,158)(105,157,150,112)(106,111,151,156)(107,155,152,110)(108,109,153,154)(117,145,142,120)(118,119,143,144)>;
G:=Group( (1,126,32,93)(2,94,33,127)(3,128,34,95)(4,96,35,129)(5,130,36,97)(6,98,37,131)(7,132,38,99)(8,100,39,133)(9,134,40,81)(10,82,21,135)(11,136,22,83)(12,84,23,137)(13,138,24,85)(14,86,25,139)(15,140,26,87)(16,88,27,121)(17,122,28,89)(18,90,29,123)(19,124,30,91)(20,92,31,125)(41,105,69,160)(42,141,70,106)(43,107,71,142)(44,143,72,108)(45,109,73,144)(46,145,74,110)(47,111,75,146)(48,147,76,112)(49,113,77,148)(50,149,78,114)(51,115,79,150)(52,151,80,116)(53,117,61,152)(54,153,62,118)(55,119,63,154)(56,155,64,120)(57,101,65,156)(58,157,66,102)(59,103,67,158)(60,159,68,104), (1,50,11,60)(2,41,12,51)(3,52,13,42)(4,43,14,53)(5,54,15,44)(6,45,16,55)(7,56,17,46)(8,47,18,57)(9,58,19,48)(10,49,20,59)(21,77,31,67)(22,68,32,78)(23,79,33,69)(24,70,34,80)(25,61,35,71)(26,72,36,62)(27,63,37,73)(28,74,38,64)(29,65,39,75)(30,76,40,66)(81,102,91,112)(82,113,92,103)(83,104,93,114)(84,115,94,105)(85,106,95,116)(86,117,96,107)(87,108,97,118)(88,119,98,109)(89,110,99,120)(90,101,100,111)(121,154,131,144)(122,145,132,155)(123,156,133,146)(124,147,134,157)(125,158,135,148)(126,149,136,159)(127,160,137,150)(128,151,138,141)(129,142,139,152)(130,153,140,143), (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), (1,10,22,31)(2,30,23,9)(3,8,24,29)(4,28,25,7)(5,6,26,27)(11,20,32,21)(12,40,33,19)(13,18,34,39)(14,38,35,17)(15,16,36,37)(41,66,79,48)(42,47,80,65)(43,64,61,46)(44,45,62,63)(49,78,67,60)(50,59,68,77)(51,76,69,58)(52,57,70,75)(53,74,71,56)(54,55,72,73)(81,94,124,137)(82,136,125,93)(83,92,126,135)(84,134,127,91)(85,90,128,133)(86,132,129,89)(87,88,130,131)(95,100,138,123)(96,122,139,99)(97,98,140,121)(101,141,146,116)(102,115,147,160)(103,159,148,114)(104,113,149,158)(105,157,150,112)(106,111,151,156)(107,155,152,110)(108,109,153,154)(117,145,142,120)(118,119,143,144) );
G=PermutationGroup([[(1,126,32,93),(2,94,33,127),(3,128,34,95),(4,96,35,129),(5,130,36,97),(6,98,37,131),(7,132,38,99),(8,100,39,133),(9,134,40,81),(10,82,21,135),(11,136,22,83),(12,84,23,137),(13,138,24,85),(14,86,25,139),(15,140,26,87),(16,88,27,121),(17,122,28,89),(18,90,29,123),(19,124,30,91),(20,92,31,125),(41,105,69,160),(42,141,70,106),(43,107,71,142),(44,143,72,108),(45,109,73,144),(46,145,74,110),(47,111,75,146),(48,147,76,112),(49,113,77,148),(50,149,78,114),(51,115,79,150),(52,151,80,116),(53,117,61,152),(54,153,62,118),(55,119,63,154),(56,155,64,120),(57,101,65,156),(58,157,66,102),(59,103,67,158),(60,159,68,104)], [(1,50,11,60),(2,41,12,51),(3,52,13,42),(4,43,14,53),(5,54,15,44),(6,45,16,55),(7,56,17,46),(8,47,18,57),(9,58,19,48),(10,49,20,59),(21,77,31,67),(22,68,32,78),(23,79,33,69),(24,70,34,80),(25,61,35,71),(26,72,36,62),(27,63,37,73),(28,74,38,64),(29,65,39,75),(30,76,40,66),(81,102,91,112),(82,113,92,103),(83,104,93,114),(84,115,94,105),(85,106,95,116),(86,117,96,107),(87,108,97,118),(88,119,98,109),(89,110,99,120),(90,101,100,111),(121,154,131,144),(122,145,132,155),(123,156,133,146),(124,147,134,157),(125,158,135,148),(126,149,136,159),(127,160,137,150),(128,151,138,141),(129,142,139,152),(130,153,140,143)], [(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)], [(1,10,22,31),(2,30,23,9),(3,8,24,29),(4,28,25,7),(5,6,26,27),(11,20,32,21),(12,40,33,19),(13,18,34,39),(14,38,35,17),(15,16,36,37),(41,66,79,48),(42,47,80,65),(43,64,61,46),(44,45,62,63),(49,78,67,60),(50,59,68,77),(51,76,69,58),(52,57,70,75),(53,74,71,56),(54,55,72,73),(81,94,124,137),(82,136,125,93),(83,92,126,135),(84,134,127,91),(85,90,128,133),(86,132,129,89),(87,88,130,131),(95,100,138,123),(96,122,139,99),(97,98,140,121),(101,141,146,116),(102,115,147,160),(103,159,148,114),(104,113,149,158),(105,157,150,112),(106,111,151,156),(107,155,152,110),(108,109,153,154),(117,145,142,120),(118,119,143,144)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 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 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | C42.241D10 | C20⋊2Q8 | D5×C42 | Dic5⋊3Q8 | C4.Dic10 | C4⋊C4⋊7D5 | D10⋊2Q8 | Q8×Dic5 | D10⋊3Q8 | C5×C4⋊Q8 | C4×D5 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 8 | 2 | 8 | 4 | 4 | 4 | 4 |
Matrix representation of C42.241D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 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 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
C42.241D10 in GAP, Magma, Sage, TeX
C_4^2._{241}D_{10} % in TeX
G:=Group("C4^2.241D10"); // GroupNames label
G:=SmallGroup(320,1400);
// by ID
G=gap.SmallGroup(320,1400);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,570,185,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^9>;
// generators/relations