metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.125D10, C10.92- 1+4, (C4×Q8)⋊6D5, (Q8×D5)⋊5C4, (Q8×C20)⋊7C2, (Q8×Dic5)⋊8C2, Q8.12(C4×D5), C4⋊C4.323D10, (C4×Dic10)⋊38C2, C10.46(C23×C4), C20.70(C22×C4), (C2×Q8).200D10, C42⋊D5.3C2, Dic5⋊3Q8⋊18C2, (C2×C10).116C24, (C2×C20).495C23, (C4×C20).168C22, Dic10.35(C2×C4), D10.41(C22×C4), C22.35(C23×D5), C4⋊Dic5.366C22, (Q8×C10).216C22, Dic5.19(C22×C4), (C4×Dic5).92C22, C2.4(D4.10D10), C2.2(Q8.10D10), C5⋊3(C23.32C23), (C2×Dic5).222C23, (C22×D5).185C23, D10⋊C4.124C22, (C2×Dic10).298C22, C10.D4.137C22, C4.35(C2×C4×D5), (C2×Q8×D5).6C2, (C4×D5).9(C2×C4), C2.27(D5×C22×C4), (C5×Q8).31(C2×C4), (C2×C4×D5).78C22, C4⋊C4⋊7D5.10C2, (C5×C4⋊C4).344C22, (C2×C4).288(C22×D5), SmallGroup(320,1244)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.125D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >
Subgroups: 718 in 266 conjugacy classes, 151 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, 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, C42⋊C2, C4×Q8, C4×Q8, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.32C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, C4×Dic10, C42⋊D5, Dic5⋊3Q8, C4⋊C4⋊7D5, Q8×Dic5, Q8×C20, C2×Q8×D5, C42.125D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2- 1+4, C4×D5, C22×D5, C23.32C23, C2×C4×D5, C23×D5, D5×C22×C4, Q8.10D10, D4.10D10, C42.125D10
►On 160 points
Generators in S160
(1 40 11 30)(2 31 12 21)(3 22 13 32)(4 33 14 23)(5 24 15 34)(6 35 16 25)(7 26 17 36)(8 37 18 27)(9 28 19 38)(10 39 20 29)(41 79 51 69)(42 70 52 80)(43 61 53 71)(44 72 54 62)(45 63 55 73)(46 74 56 64)(47 65 57 75)(48 76 58 66)(49 67 59 77)(50 78 60 68)(81 104 91 114)(82 115 92 105)(83 106 93 116)(84 117 94 107)(85 108 95 118)(86 119 96 109)(87 110 97 120)(88 101 98 111)(89 112 99 102)(90 103 100 113)(121 153 131 143)(122 144 132 154)(123 155 133 145)(124 146 134 156)(125 157 135 147)(126 148 136 158)(127 159 137 149)(128 150 138 160)(129 141 139 151)(130 152 140 142)
(1 60 100 123)(2 41 81 124)(3 42 82 125)(4 43 83 126)(5 44 84 127)(6 45 85 128)(7 46 86 129)(8 47 87 130)(9 48 88 131)(10 49 89 132)(11 50 90 133)(12 51 91 134)(13 52 92 135)(14 53 93 136)(15 54 94 137)(16 55 95 138)(17 56 96 139)(18 57 97 140)(19 58 98 121)(20 59 99 122)(21 69 114 156)(22 70 115 157)(23 71 116 158)(24 72 117 159)(25 73 118 160)(26 74 119 141)(27 75 120 142)(28 76 101 143)(29 77 102 144)(30 78 103 145)(31 79 104 146)(32 80 105 147)(33 61 106 148)(34 62 107 149)(35 63 108 150)(36 64 109 151)(37 65 110 152)(38 66 111 153)(39 67 112 154)(40 68 113 155)
(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 38 31 28)(22 27 32 37)(23 36 33 26)(24 25 34 35)(29 30 39 40)(41 48 51 58)(42 57 52 47)(43 46 53 56)(44 55 54 45)(49 60 59 50)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)(81 98 91 88)(82 87 92 97)(83 96 93 86)(84 85 94 95)(89 90 99 100)(101 114 111 104)(102 103 112 113)(105 110 115 120)(106 119 116 109)(107 108 117 118)(121 124 131 134)(122 133 132 123)(125 140 135 130)(126 129 136 139)(127 138 137 128)(141 148 151 158)(142 157 152 147)(143 146 153 156)(144 155 154 145)(149 160 159 150)
G:=sub<Sym(160)| (1,40,11,30)(2,31,12,21)(3,22,13,32)(4,33,14,23)(5,24,15,34)(6,35,16,25)(7,26,17,36)(8,37,18,27)(9,28,19,38)(10,39,20,29)(41,79,51,69)(42,70,52,80)(43,61,53,71)(44,72,54,62)(45,63,55,73)(46,74,56,64)(47,65,57,75)(48,76,58,66)(49,67,59,77)(50,78,60,68)(81,104,91,114)(82,115,92,105)(83,106,93,116)(84,117,94,107)(85,108,95,118)(86,119,96,109)(87,110,97,120)(88,101,98,111)(89,112,99,102)(90,103,100,113)(121,153,131,143)(122,144,132,154)(123,155,133,145)(124,146,134,156)(125,157,135,147)(126,148,136,158)(127,159,137,149)(128,150,138,160)(129,141,139,151)(130,152,140,142), (1,60,100,123)(2,41,81,124)(3,42,82,125)(4,43,83,126)(5,44,84,127)(6,45,85,128)(7,46,86,129)(8,47,87,130)(9,48,88,131)(10,49,89,132)(11,50,90,133)(12,51,91,134)(13,52,92,135)(14,53,93,136)(15,54,94,137)(16,55,95,138)(17,56,96,139)(18,57,97,140)(19,58,98,121)(20,59,99,122)(21,69,114,156)(22,70,115,157)(23,71,116,158)(24,72,117,159)(25,73,118,160)(26,74,119,141)(27,75,120,142)(28,76,101,143)(29,77,102,144)(30,78,103,145)(31,79,104,146)(32,80,105,147)(33,61,106,148)(34,62,107,149)(35,63,108,150)(36,64,109,151)(37,65,110,152)(38,66,111,153)(39,67,112,154)(40,68,113,155), (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)(21,38,31,28)(22,27,32,37)(23,36,33,26)(24,25,34,35)(29,30,39,40)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)(81,98,91,88)(82,87,92,97)(83,96,93,86)(84,85,94,95)(89,90,99,100)(101,114,111,104)(102,103,112,113)(105,110,115,120)(106,119,116,109)(107,108,117,118)(121,124,131,134)(122,133,132,123)(125,140,135,130)(126,129,136,139)(127,138,137,128)(141,148,151,158)(142,157,152,147)(143,146,153,156)(144,155,154,145)(149,160,159,150)>;
G:=Group( (1,40,11,30)(2,31,12,21)(3,22,13,32)(4,33,14,23)(5,24,15,34)(6,35,16,25)(7,26,17,36)(8,37,18,27)(9,28,19,38)(10,39,20,29)(41,79,51,69)(42,70,52,80)(43,61,53,71)(44,72,54,62)(45,63,55,73)(46,74,56,64)(47,65,57,75)(48,76,58,66)(49,67,59,77)(50,78,60,68)(81,104,91,114)(82,115,92,105)(83,106,93,116)(84,117,94,107)(85,108,95,118)(86,119,96,109)(87,110,97,120)(88,101,98,111)(89,112,99,102)(90,103,100,113)(121,153,131,143)(122,144,132,154)(123,155,133,145)(124,146,134,156)(125,157,135,147)(126,148,136,158)(127,159,137,149)(128,150,138,160)(129,141,139,151)(130,152,140,142), (1,60,100,123)(2,41,81,124)(3,42,82,125)(4,43,83,126)(5,44,84,127)(6,45,85,128)(7,46,86,129)(8,47,87,130)(9,48,88,131)(10,49,89,132)(11,50,90,133)(12,51,91,134)(13,52,92,135)(14,53,93,136)(15,54,94,137)(16,55,95,138)(17,56,96,139)(18,57,97,140)(19,58,98,121)(20,59,99,122)(21,69,114,156)(22,70,115,157)(23,71,116,158)(24,72,117,159)(25,73,118,160)(26,74,119,141)(27,75,120,142)(28,76,101,143)(29,77,102,144)(30,78,103,145)(31,79,104,146)(32,80,105,147)(33,61,106,148)(34,62,107,149)(35,63,108,150)(36,64,109,151)(37,65,110,152)(38,66,111,153)(39,67,112,154)(40,68,113,155), (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)(21,38,31,28)(22,27,32,37)(23,36,33,26)(24,25,34,35)(29,30,39,40)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)(81,98,91,88)(82,87,92,97)(83,96,93,86)(84,85,94,95)(89,90,99,100)(101,114,111,104)(102,103,112,113)(105,110,115,120)(106,119,116,109)(107,108,117,118)(121,124,131,134)(122,133,132,123)(125,140,135,130)(126,129,136,139)(127,138,137,128)(141,148,151,158)(142,157,152,147)(143,146,153,156)(144,155,154,145)(149,160,159,150) );
G=PermutationGroup([[(1,40,11,30),(2,31,12,21),(3,22,13,32),(4,33,14,23),(5,24,15,34),(6,35,16,25),(7,26,17,36),(8,37,18,27),(9,28,19,38),(10,39,20,29),(41,79,51,69),(42,70,52,80),(43,61,53,71),(44,72,54,62),(45,63,55,73),(46,74,56,64),(47,65,57,75),(48,76,58,66),(49,67,59,77),(50,78,60,68),(81,104,91,114),(82,115,92,105),(83,106,93,116),(84,117,94,107),(85,108,95,118),(86,119,96,109),(87,110,97,120),(88,101,98,111),(89,112,99,102),(90,103,100,113),(121,153,131,143),(122,144,132,154),(123,155,133,145),(124,146,134,156),(125,157,135,147),(126,148,136,158),(127,159,137,149),(128,150,138,160),(129,141,139,151),(130,152,140,142)], [(1,60,100,123),(2,41,81,124),(3,42,82,125),(4,43,83,126),(5,44,84,127),(6,45,85,128),(7,46,86,129),(8,47,87,130),(9,48,88,131),(10,49,89,132),(11,50,90,133),(12,51,91,134),(13,52,92,135),(14,53,93,136),(15,54,94,137),(16,55,95,138),(17,56,96,139),(18,57,97,140),(19,58,98,121),(20,59,99,122),(21,69,114,156),(22,70,115,157),(23,71,116,158),(24,72,117,159),(25,73,118,160),(26,74,119,141),(27,75,120,142),(28,76,101,143),(29,77,102,144),(30,78,103,145),(31,79,104,146),(32,80,105,147),(33,61,106,148),(34,62,107,149),(35,63,108,150),(36,64,109,151),(37,65,110,152),(38,66,111,153),(39,67,112,154),(40,68,113,155)], [(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),(21,38,31,28),(22,27,32,37),(23,36,33,26),(24,25,34,35),(29,30,39,40),(41,48,51,58),(42,57,52,47),(43,46,53,56),(44,55,54,45),(49,60,59,50),(61,64,71,74),(62,73,72,63),(65,80,75,70),(66,69,76,79),(67,78,77,68),(81,98,91,88),(82,87,92,97),(83,96,93,86),(84,85,94,95),(89,90,99,100),(101,114,111,104),(102,103,112,113),(105,110,115,120),(106,119,116,109),(107,108,117,118),(121,124,131,134),(122,133,132,123),(125,140,135,130),(126,129,136,139),(127,138,137,128),(141,148,151,158),(142,157,152,147),(143,146,153,156),(144,155,154,145),(149,160,159,150)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4N | 4O | ··· | 4AB | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | C4×D5 | 2- 1+4 | Q8.10D10 | D4.10D10 |
| kernel | C42.125D10 | C4×Dic10 | C42⋊D5 | Dic5⋊3Q8 | C4⋊C4⋊7D5 | Q8×Dic5 | Q8×C20 | C2×Q8×D5 | Q8×D5 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 2 | 6 | 6 | 2 | 16 | 2 | 4 | 4 |
Matrix representation of C42.125D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 1 | 0 |
| 0 | 0 | 0 | 11 | 0 | 1 |
| 0 | 0 | 1 | 0 | 30 | 0 |
| 0 | 0 | 0 | 1 | 0 | 30 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,0,1,0,0,0,0,11,0,1,0,0,1,0,30,0,0,0,0,1,0,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,40,34,0,0,0,0,7,7,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,7,7,0,0,0,0,40,34,0,0] >;
C42.125D10 in GAP, Magma, Sage, TeX
C_4^2._{125}D_{10} % in TeX
G:=Group("C4^2.125D10"); // GroupNames label
G:=SmallGroup(320,1244);
// by ID
G=gap.SmallGroup(320,1244);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,1123,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations