metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.213D10, (C2×D4).46D10, (C2×C20).270D4, (C2×Q8).36D10, C4.4D4.6D5, C20.67(C4○D4), Q8⋊Dic5⋊21C2, C20.6Q8⋊12C2, C10.104(C4○D8), C4.21(D4⋊2D5), (C2×C20).374C23, (C4×C20).105C22, D4⋊Dic5.13C2, (D4×C10).62C22, (Q8×C10).54C22, C2.9(C20.17D4), C10.42(C4.4D4), C4⋊Dic5.151C22, C2.23(D4.8D10), C5⋊4(C42.78C22), (C4×C5⋊2C8)⋊11C2, (C2×C10).505(C2×D4), (C5×C4.4D4).4C2, (C2×C4).109(C5⋊D4), (C2×C4).474(C22×D5), C22.180(C2×C5⋊D4), (C2×C5⋊2C8).260C22, SmallGroup(320,683)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.213D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 302 in 96 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.78C22, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C4×C5⋊2C8, D4⋊Dic5, Q8⋊Dic5, C20.6Q8, C5×C4.4D4, C42.213D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C4○D8, C5⋊D4, C22×D5, C42.78C22, D4⋊2D5, C2×C5⋊D4, C20.17D4, D4.8D10, C42.213D10
►On 160 points
Generators in S160
(1 68 11 52)(2 64 12 58)(3 70 13 54)(4 66 14 60)(5 62 15 56)(6 69 19 53)(7 65 20 59)(8 61 16 55)(9 67 17 51)(10 63 18 57)(21 98 125 88)(22 101 126 111)(23 100 127 90)(24 103 128 113)(25 92 129 82)(26 105 130 115)(27 94 121 84)(28 107 122 117)(29 96 123 86)(30 109 124 119)(31 50 153 45)(32 77 154 72)(33 42 155 47)(34 79 156 74)(35 44 157 49)(36 71 158 76)(37 46 159 41)(38 73 160 78)(39 48 151 43)(40 75 152 80)(81 133 91 145)(83 135 93 147)(85 137 95 149)(87 139 97 141)(89 131 99 143)(102 144 112 132)(104 146 114 134)(106 148 116 136)(108 150 118 138)(110 142 120 140)
(1 152 18 157)(2 158 19 153)(3 154 20 159)(4 160 16 155)(5 156 17 151)(6 31 12 36)(7 37 13 32)(8 33 14 38)(9 39 15 34)(10 35 11 40)(21 26 142 147)(22 148 143 27)(23 28 144 149)(24 150 145 29)(25 30 146 141)(41 70 72 59)(42 60 73 61)(43 62 74 51)(44 52 75 63)(45 64 76 53)(46 54 77 65)(47 66 78 55)(48 56 79 67)(49 68 80 57)(50 58 71 69)(81 96 103 118)(82 119 104 97)(83 98 105 120)(84 111 106 99)(85 100 107 112)(86 113 108 91)(87 92 109 114)(88 115 110 93)(89 94 101 116)(90 117 102 95)(121 126 136 131)(122 132 137 127)(123 128 138 133)(124 134 139 129)(125 130 140 135)
(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)
(1 109 11 119)(2 103 12 113)(3 107 13 117)(4 101 14 111)(5 105 15 115)(6 91 19 81)(7 95 20 85)(8 99 16 89)(9 93 17 83)(10 97 18 87)(21 48 125 43)(22 61 126 55)(23 46 127 41)(24 69 128 53)(25 44 129 49)(26 67 130 51)(27 42 121 47)(28 65 122 59)(29 50 123 45)(30 63 124 57)(31 108 153 118)(32 90 154 100)(33 106 155 116)(34 88 156 98)(35 104 157 114)(36 86 158 96)(37 102 159 112)(38 84 160 94)(39 110 151 120)(40 82 152 92)(52 139 68 141)(54 137 70 149)(56 135 62 147)(58 133 64 145)(60 131 66 143)(71 138 76 150)(72 144 77 132)(73 136 78 148)(74 142 79 140)(75 134 80 146)
G:=sub<Sym(160)| (1,68,11,52)(2,64,12,58)(3,70,13,54)(4,66,14,60)(5,62,15,56)(6,69,19,53)(7,65,20,59)(8,61,16,55)(9,67,17,51)(10,63,18,57)(21,98,125,88)(22,101,126,111)(23,100,127,90)(24,103,128,113)(25,92,129,82)(26,105,130,115)(27,94,121,84)(28,107,122,117)(29,96,123,86)(30,109,124,119)(31,50,153,45)(32,77,154,72)(33,42,155,47)(34,79,156,74)(35,44,157,49)(36,71,158,76)(37,46,159,41)(38,73,160,78)(39,48,151,43)(40,75,152,80)(81,133,91,145)(83,135,93,147)(85,137,95,149)(87,139,97,141)(89,131,99,143)(102,144,112,132)(104,146,114,134)(106,148,116,136)(108,150,118,138)(110,142,120,140), (1,152,18,157)(2,158,19,153)(3,154,20,159)(4,160,16,155)(5,156,17,151)(6,31,12,36)(7,37,13,32)(8,33,14,38)(9,39,15,34)(10,35,11,40)(21,26,142,147)(22,148,143,27)(23,28,144,149)(24,150,145,29)(25,30,146,141)(41,70,72,59)(42,60,73,61)(43,62,74,51)(44,52,75,63)(45,64,76,53)(46,54,77,65)(47,66,78,55)(48,56,79,67)(49,68,80,57)(50,58,71,69)(81,96,103,118)(82,119,104,97)(83,98,105,120)(84,111,106,99)(85,100,107,112)(86,113,108,91)(87,92,109,114)(88,115,110,93)(89,94,101,116)(90,117,102,95)(121,126,136,131)(122,132,137,127)(123,128,138,133)(124,134,139,129)(125,130,140,135), (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), (1,109,11,119)(2,103,12,113)(3,107,13,117)(4,101,14,111)(5,105,15,115)(6,91,19,81)(7,95,20,85)(8,99,16,89)(9,93,17,83)(10,97,18,87)(21,48,125,43)(22,61,126,55)(23,46,127,41)(24,69,128,53)(25,44,129,49)(26,67,130,51)(27,42,121,47)(28,65,122,59)(29,50,123,45)(30,63,124,57)(31,108,153,118)(32,90,154,100)(33,106,155,116)(34,88,156,98)(35,104,157,114)(36,86,158,96)(37,102,159,112)(38,84,160,94)(39,110,151,120)(40,82,152,92)(52,139,68,141)(54,137,70,149)(56,135,62,147)(58,133,64,145)(60,131,66,143)(71,138,76,150)(72,144,77,132)(73,136,78,148)(74,142,79,140)(75,134,80,146)>;
G:=Group( (1,68,11,52)(2,64,12,58)(3,70,13,54)(4,66,14,60)(5,62,15,56)(6,69,19,53)(7,65,20,59)(8,61,16,55)(9,67,17,51)(10,63,18,57)(21,98,125,88)(22,101,126,111)(23,100,127,90)(24,103,128,113)(25,92,129,82)(26,105,130,115)(27,94,121,84)(28,107,122,117)(29,96,123,86)(30,109,124,119)(31,50,153,45)(32,77,154,72)(33,42,155,47)(34,79,156,74)(35,44,157,49)(36,71,158,76)(37,46,159,41)(38,73,160,78)(39,48,151,43)(40,75,152,80)(81,133,91,145)(83,135,93,147)(85,137,95,149)(87,139,97,141)(89,131,99,143)(102,144,112,132)(104,146,114,134)(106,148,116,136)(108,150,118,138)(110,142,120,140), (1,152,18,157)(2,158,19,153)(3,154,20,159)(4,160,16,155)(5,156,17,151)(6,31,12,36)(7,37,13,32)(8,33,14,38)(9,39,15,34)(10,35,11,40)(21,26,142,147)(22,148,143,27)(23,28,144,149)(24,150,145,29)(25,30,146,141)(41,70,72,59)(42,60,73,61)(43,62,74,51)(44,52,75,63)(45,64,76,53)(46,54,77,65)(47,66,78,55)(48,56,79,67)(49,68,80,57)(50,58,71,69)(81,96,103,118)(82,119,104,97)(83,98,105,120)(84,111,106,99)(85,100,107,112)(86,113,108,91)(87,92,109,114)(88,115,110,93)(89,94,101,116)(90,117,102,95)(121,126,136,131)(122,132,137,127)(123,128,138,133)(124,134,139,129)(125,130,140,135), (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), (1,109,11,119)(2,103,12,113)(3,107,13,117)(4,101,14,111)(5,105,15,115)(6,91,19,81)(7,95,20,85)(8,99,16,89)(9,93,17,83)(10,97,18,87)(21,48,125,43)(22,61,126,55)(23,46,127,41)(24,69,128,53)(25,44,129,49)(26,67,130,51)(27,42,121,47)(28,65,122,59)(29,50,123,45)(30,63,124,57)(31,108,153,118)(32,90,154,100)(33,106,155,116)(34,88,156,98)(35,104,157,114)(36,86,158,96)(37,102,159,112)(38,84,160,94)(39,110,151,120)(40,82,152,92)(52,139,68,141)(54,137,70,149)(56,135,62,147)(58,133,64,145)(60,131,66,143)(71,138,76,150)(72,144,77,132)(73,136,78,148)(74,142,79,140)(75,134,80,146) );
G=PermutationGroup([[(1,68,11,52),(2,64,12,58),(3,70,13,54),(4,66,14,60),(5,62,15,56),(6,69,19,53),(7,65,20,59),(8,61,16,55),(9,67,17,51),(10,63,18,57),(21,98,125,88),(22,101,126,111),(23,100,127,90),(24,103,128,113),(25,92,129,82),(26,105,130,115),(27,94,121,84),(28,107,122,117),(29,96,123,86),(30,109,124,119),(31,50,153,45),(32,77,154,72),(33,42,155,47),(34,79,156,74),(35,44,157,49),(36,71,158,76),(37,46,159,41),(38,73,160,78),(39,48,151,43),(40,75,152,80),(81,133,91,145),(83,135,93,147),(85,137,95,149),(87,139,97,141),(89,131,99,143),(102,144,112,132),(104,146,114,134),(106,148,116,136),(108,150,118,138),(110,142,120,140)], [(1,152,18,157),(2,158,19,153),(3,154,20,159),(4,160,16,155),(5,156,17,151),(6,31,12,36),(7,37,13,32),(8,33,14,38),(9,39,15,34),(10,35,11,40),(21,26,142,147),(22,148,143,27),(23,28,144,149),(24,150,145,29),(25,30,146,141),(41,70,72,59),(42,60,73,61),(43,62,74,51),(44,52,75,63),(45,64,76,53),(46,54,77,65),(47,66,78,55),(48,56,79,67),(49,68,80,57),(50,58,71,69),(81,96,103,118),(82,119,104,97),(83,98,105,120),(84,111,106,99),(85,100,107,112),(86,113,108,91),(87,92,109,114),(88,115,110,93),(89,94,101,116),(90,117,102,95),(121,126,136,131),(122,132,137,127),(123,128,138,133),(124,134,139,129),(125,130,140,135)], [(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)], [(1,109,11,119),(2,103,12,113),(3,107,13,117),(4,101,14,111),(5,105,15,115),(6,91,19,81),(7,95,20,85),(8,99,16,89),(9,93,17,83),(10,97,18,87),(21,48,125,43),(22,61,126,55),(23,46,127,41),(24,69,128,53),(25,44,129,49),(26,67,130,51),(27,42,121,47),(28,65,122,59),(29,50,123,45),(30,63,124,57),(31,108,153,118),(32,90,154,100),(33,106,155,116),(34,88,156,98),(35,104,157,114),(36,86,158,96),(37,102,159,112),(38,84,160,94),(39,110,151,120),(40,82,152,92),(52,139,68,141),(54,137,70,149),(56,135,62,147),(58,133,64,145),(60,131,66,143),(71,138,76,150),(72,144,77,132),(73,136,78,148),(74,142,79,140),(75,134,80,146)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | ··· | 2 | 8 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | D4⋊2D5 | D4.8D10 |
| kernel | C42.213D10 | C4×C5⋊2C8 | D4⋊Dic5 | Q8⋊Dic5 | C20.6Q8 | C5×C4.4D4 | C2×C20 | C4.4D4 | C20 | C42 | C2×D4 | C2×Q8 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 4 | 8 |
Matrix representation of C42.213D10 ►in GL6(𝔽41)
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 38 | 31 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 29 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 15 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 22 |
| 0 | 0 | 0 | 0 | 2 | 30 |
G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,37,38,0,0,0,0,0,31],[12,29,0,0,0,0,29,29,0,0,0,0,0,0,26,15,0,0,0,0,15,15,0,0,0,0,0,0,11,2,0,0,0,0,22,30] >;
C42.213D10 in GAP, Magma, Sage, TeX
C_4^2._{213}D_{10} % in TeX
G:=Group("C4^2.213D10"); // GroupNames label
G:=SmallGroup(320,683);
// by ID
G=gap.SmallGroup(320,683);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,590,471,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations