metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.135D10, C10.672- 1+4, (C4×Q8)⋊17D5, (Q8×C20)⋊19C2, C4⋊C4.302D10, C20⋊2Q8⋊29C2, (C4×D20).23C2, D10⋊3Q8⋊11C2, (C4×Dic10)⋊41C2, C4.69(C4○D20), (C2×Q8).183D10, C20.122(C4○D4), (C4×C20).180C22, (C2×C10).128C24, (C2×C20).591C23, C4.50(Q8⋊2D5), C4.D20.13C2, (C2×D20).227C22, C4⋊Dic5.400C22, (Q8×C10).228C22, (C22×D5).50C23, C22.149(C23×D5), D10⋊C4.56C22, C5⋊4(C22.50C24), (C4×Dic5).230C22, (C2×Dic5).228C23, C2.25(D4.10D10), (C2×Dic10).251C22, C10.D4.157C22, C4⋊C4⋊D5⋊11C2, C4⋊C4⋊7D5⋊18C2, C2.67(C2×C4○D20), (C2×C4×D5).87C22, C10.113(C2×C4○D4), C2.13(C2×Q8⋊2D5), (C5×C4⋊C4).356C22, (C2×C4).172(C22×D5), SmallGroup(320,1256)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.135D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c9 >
Subgroups: 694 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.50C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C4×Dic10, C20⋊2Q8, C4×D20, C4.D20, C4⋊C4⋊7D5, C4⋊C4⋊D5, D10⋊3Q8, Q8×C20, C42.135D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, C4○D20, Q8⋊2D5, C23×D5, C2×C4○D20, C2×Q8⋊2D5, D4.10D10, C42.135D10
►On 160 points
Generators in S160
(1 60 66 29)(2 41 67 30)(3 42 68 31)(4 43 69 32)(5 44 70 33)(6 45 71 34)(7 46 72 35)(8 47 73 36)(9 48 74 37)(10 49 75 38)(11 50 76 39)(12 51 77 40)(13 52 78 21)(14 53 79 22)(15 54 80 23)(16 55 61 24)(17 56 62 25)(18 57 63 26)(19 58 64 27)(20 59 65 28)(81 111 133 149)(82 112 134 150)(83 113 135 151)(84 114 136 152)(85 115 137 153)(86 116 138 154)(87 117 139 155)(88 118 140 156)(89 119 121 157)(90 120 122 158)(91 101 123 159)(92 102 124 160)(93 103 125 141)(94 104 126 142)(95 105 127 143)(96 106 128 144)(97 107 129 145)(98 108 130 146)(99 109 131 147)(100 110 132 148)
(1 93 76 135)(2 84 77 126)(3 95 78 137)(4 86 79 128)(5 97 80 139)(6 88 61 130)(7 99 62 121)(8 90 63 132)(9 81 64 123)(10 92 65 134)(11 83 66 125)(12 94 67 136)(13 85 68 127)(14 96 69 138)(15 87 70 129)(16 98 71 140)(17 89 72 131)(18 100 73 122)(19 91 74 133)(20 82 75 124)(21 153 42 105)(22 144 43 116)(23 155 44 107)(24 146 45 118)(25 157 46 109)(26 148 47 120)(27 159 48 111)(28 150 49 102)(29 141 50 113)(30 152 51 104)(31 143 52 115)(32 154 53 106)(33 145 54 117)(34 156 55 108)(35 147 56 119)(36 158 57 110)(37 149 58 101)(38 160 59 112)(39 151 60 103)(40 142 41 114)
(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 57 31 47)(22 46 32 56)(23 55 33 45)(24 44 34 54)(25 53 35 43)(26 42 36 52)(27 51 37 41)(28 60 38 50)(29 49 39 59)(30 58 40 48)(61 70 71 80)(62 79 72 69)(63 68 73 78)(64 77 74 67)(65 66 75 76)(81 84 91 94)(82 93 92 83)(85 100 95 90)(86 89 96 99)(87 98 97 88)(101 142 111 152)(102 151 112 141)(103 160 113 150)(104 149 114 159)(105 158 115 148)(106 147 116 157)(107 156 117 146)(108 145 118 155)(109 154 119 144)(110 143 120 153)(121 128 131 138)(122 137 132 127)(123 126 133 136)(124 135 134 125)(129 140 139 130)
G:=sub<Sym(160)| (1,60,66,29)(2,41,67,30)(3,42,68,31)(4,43,69,32)(5,44,70,33)(6,45,71,34)(7,46,72,35)(8,47,73,36)(9,48,74,37)(10,49,75,38)(11,50,76,39)(12,51,77,40)(13,52,78,21)(14,53,79,22)(15,54,80,23)(16,55,61,24)(17,56,62,25)(18,57,63,26)(19,58,64,27)(20,59,65,28)(81,111,133,149)(82,112,134,150)(83,113,135,151)(84,114,136,152)(85,115,137,153)(86,116,138,154)(87,117,139,155)(88,118,140,156)(89,119,121,157)(90,120,122,158)(91,101,123,159)(92,102,124,160)(93,103,125,141)(94,104,126,142)(95,105,127,143)(96,106,128,144)(97,107,129,145)(98,108,130,146)(99,109,131,147)(100,110,132,148), (1,93,76,135)(2,84,77,126)(3,95,78,137)(4,86,79,128)(5,97,80,139)(6,88,61,130)(7,99,62,121)(8,90,63,132)(9,81,64,123)(10,92,65,134)(11,83,66,125)(12,94,67,136)(13,85,68,127)(14,96,69,138)(15,87,70,129)(16,98,71,140)(17,89,72,131)(18,100,73,122)(19,91,74,133)(20,82,75,124)(21,153,42,105)(22,144,43,116)(23,155,44,107)(24,146,45,118)(25,157,46,109)(26,148,47,120)(27,159,48,111)(28,150,49,102)(29,141,50,113)(30,152,51,104)(31,143,52,115)(32,154,53,106)(33,145,54,117)(34,156,55,108)(35,147,56,119)(36,158,57,110)(37,149,58,101)(38,160,59,112)(39,151,60,103)(40,142,41,114), (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)(21,57,31,47)(22,46,32,56)(23,55,33,45)(24,44,34,54)(25,53,35,43)(26,42,36,52)(27,51,37,41)(28,60,38,50)(29,49,39,59)(30,58,40,48)(61,70,71,80)(62,79,72,69)(63,68,73,78)(64,77,74,67)(65,66,75,76)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,142,111,152)(102,151,112,141)(103,160,113,150)(104,149,114,159)(105,158,115,148)(106,147,116,157)(107,156,117,146)(108,145,118,155)(109,154,119,144)(110,143,120,153)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130)>;
G:=Group( (1,60,66,29)(2,41,67,30)(3,42,68,31)(4,43,69,32)(5,44,70,33)(6,45,71,34)(7,46,72,35)(8,47,73,36)(9,48,74,37)(10,49,75,38)(11,50,76,39)(12,51,77,40)(13,52,78,21)(14,53,79,22)(15,54,80,23)(16,55,61,24)(17,56,62,25)(18,57,63,26)(19,58,64,27)(20,59,65,28)(81,111,133,149)(82,112,134,150)(83,113,135,151)(84,114,136,152)(85,115,137,153)(86,116,138,154)(87,117,139,155)(88,118,140,156)(89,119,121,157)(90,120,122,158)(91,101,123,159)(92,102,124,160)(93,103,125,141)(94,104,126,142)(95,105,127,143)(96,106,128,144)(97,107,129,145)(98,108,130,146)(99,109,131,147)(100,110,132,148), (1,93,76,135)(2,84,77,126)(3,95,78,137)(4,86,79,128)(5,97,80,139)(6,88,61,130)(7,99,62,121)(8,90,63,132)(9,81,64,123)(10,92,65,134)(11,83,66,125)(12,94,67,136)(13,85,68,127)(14,96,69,138)(15,87,70,129)(16,98,71,140)(17,89,72,131)(18,100,73,122)(19,91,74,133)(20,82,75,124)(21,153,42,105)(22,144,43,116)(23,155,44,107)(24,146,45,118)(25,157,46,109)(26,148,47,120)(27,159,48,111)(28,150,49,102)(29,141,50,113)(30,152,51,104)(31,143,52,115)(32,154,53,106)(33,145,54,117)(34,156,55,108)(35,147,56,119)(36,158,57,110)(37,149,58,101)(38,160,59,112)(39,151,60,103)(40,142,41,114), (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)(21,57,31,47)(22,46,32,56)(23,55,33,45)(24,44,34,54)(25,53,35,43)(26,42,36,52)(27,51,37,41)(28,60,38,50)(29,49,39,59)(30,58,40,48)(61,70,71,80)(62,79,72,69)(63,68,73,78)(64,77,74,67)(65,66,75,76)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,142,111,152)(102,151,112,141)(103,160,113,150)(104,149,114,159)(105,158,115,148)(106,147,116,157)(107,156,117,146)(108,145,118,155)(109,154,119,144)(110,143,120,153)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130) );
G=PermutationGroup([[(1,60,66,29),(2,41,67,30),(3,42,68,31),(4,43,69,32),(5,44,70,33),(6,45,71,34),(7,46,72,35),(8,47,73,36),(9,48,74,37),(10,49,75,38),(11,50,76,39),(12,51,77,40),(13,52,78,21),(14,53,79,22),(15,54,80,23),(16,55,61,24),(17,56,62,25),(18,57,63,26),(19,58,64,27),(20,59,65,28),(81,111,133,149),(82,112,134,150),(83,113,135,151),(84,114,136,152),(85,115,137,153),(86,116,138,154),(87,117,139,155),(88,118,140,156),(89,119,121,157),(90,120,122,158),(91,101,123,159),(92,102,124,160),(93,103,125,141),(94,104,126,142),(95,105,127,143),(96,106,128,144),(97,107,129,145),(98,108,130,146),(99,109,131,147),(100,110,132,148)], [(1,93,76,135),(2,84,77,126),(3,95,78,137),(4,86,79,128),(5,97,80,139),(6,88,61,130),(7,99,62,121),(8,90,63,132),(9,81,64,123),(10,92,65,134),(11,83,66,125),(12,94,67,136),(13,85,68,127),(14,96,69,138),(15,87,70,129),(16,98,71,140),(17,89,72,131),(18,100,73,122),(19,91,74,133),(20,82,75,124),(21,153,42,105),(22,144,43,116),(23,155,44,107),(24,146,45,118),(25,157,46,109),(26,148,47,120),(27,159,48,111),(28,150,49,102),(29,141,50,113),(30,152,51,104),(31,143,52,115),(32,154,53,106),(33,145,54,117),(34,156,55,108),(35,147,56,119),(36,158,57,110),(37,149,58,101),(38,160,59,112),(39,151,60,103),(40,142,41,114)], [(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),(21,57,31,47),(22,46,32,56),(23,55,33,45),(24,44,34,54),(25,53,35,43),(26,42,36,52),(27,51,37,41),(28,60,38,50),(29,49,39,59),(30,58,40,48),(61,70,71,80),(62,79,72,69),(63,68,73,78),(64,77,74,67),(65,66,75,76),(81,84,91,94),(82,93,92,83),(85,100,95,90),(86,89,96,99),(87,98,97,88),(101,142,111,152),(102,151,112,141),(103,160,113,150),(104,149,114,159),(105,158,115,148),(106,147,116,157),(107,156,117,146),(108,145,118,155),(109,154,119,144),(110,143,120,153),(121,128,131,138),(122,137,132,127),(123,126,133,136),(124,135,134,125),(129,140,139,130)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 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 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | Q8⋊2D5 | D4.10D10 |
| kernel | C42.135D10 | C4×Dic10 | C20⋊2Q8 | C4×D20 | C4.D20 | C4⋊C4⋊7D5 | C4⋊C4⋊D5 | D10⋊3Q8 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 8 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.135D10 ►in GL4(𝔽41) generated by
| 2 | 32 | 0 | 0 |
| 37 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 23 |
| 0 | 0 | 32 | 1 |
| 35 | 34 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 40 | 32 |
| 6 | 40 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [2,37,0,0,32,39,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,40,32,0,0,23,1],[35,6,0,0,34,0,0,0,0,0,9,40,0,0,0,32],[6,35,0,0,40,35,0,0,0,0,9,0,0,0,0,9] >;
C42.135D10 in GAP, Magma, Sage, TeX
C_4^2._{135}D_{10} % in TeX
G:=Group("C4^2.135D10"); // GroupNames label
G:=SmallGroup(320,1256);
// by ID
G=gap.SmallGroup(320,1256);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations