metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.177D10, C10.392- 1+4, C4⋊Q8⋊15D5, C4⋊C4.221D10, (Q8×Dic5)⋊24C2, (C4×D20).28C2, D10⋊2Q8⋊45C2, (C4×Dic10)⋊54C2, (C2×Q8).149D10, C20.139(C4○D4), C4.19(D4⋊2D5), (C4×C20).217C22, (C2×C20).109C23, (C2×C10).276C24, C4.41(Q8⋊2D5), C20.23D4.10C2, (C2×D20).282C22, C4⋊Dic5.386C22, (Q8×C10).143C22, C22.297(C23×D5), C5⋊8(C22.50C24), (C4×Dic5).173C22, (C2×Dic5).146C23, (C22×D5).121C23, D10⋊C4.155C22, C2.40(Q8.10D10), (C2×Dic10).312C22, C10.D4.168C22, (C5×C4⋊Q8)⋊18C2, C4⋊C4⋊7D5⋊43C2, C4⋊C4⋊D5⋊46C2, C10.123(C2×C4○D4), C2.66(C2×D4⋊2D5), C2.31(C2×Q8⋊2D5), (C2×C4×D5).158C22, (C5×C4⋊C4).219C22, (C2×C4).601(C22×D5), SmallGroup(320,1404)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.177D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=b2c9 >
Subgroups: 678 in 212 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊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, 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, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C4×Dic10, C4×D20, C4⋊C4⋊7D5, D10⋊2Q8, C4⋊C4⋊D5, Q8×Dic5, C20.23D4, C5×C4⋊Q8, C42.177D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, D4⋊2D5, Q8⋊2D5, C23×D5, C2×D4⋊2D5, C2×Q8⋊2D5, Q8.10D10, C42.177D10
►On 160 points
Generators in S160
(1 149 123 61)(2 62 124 150)(3 151 125 63)(4 64 126 152)(5 153 127 65)(6 66 128 154)(7 155 129 67)(8 68 130 156)(9 157 131 69)(10 70 132 158)(11 159 133 71)(12 72 134 160)(13 141 135 73)(14 74 136 142)(15 143 137 75)(16 76 138 144)(17 145 139 77)(18 78 140 146)(19 147 121 79)(20 80 122 148)(21 59 92 118)(22 119 93 60)(23 41 94 120)(24 101 95 42)(25 43 96 102)(26 103 97 44)(27 45 98 104)(28 105 99 46)(29 47 100 106)(30 107 81 48)(31 49 82 108)(32 109 83 50)(33 51 84 110)(34 111 85 52)(35 53 86 112)(36 113 87 54)(37 55 88 114)(38 115 89 56)(39 57 90 116)(40 117 91 58)
(1 87 133 26)(2 27 134 88)(3 89 135 28)(4 29 136 90)(5 91 137 30)(6 31 138 92)(7 93 139 32)(8 33 140 94)(9 95 121 34)(10 35 122 96)(11 97 123 36)(12 37 124 98)(13 99 125 38)(14 39 126 100)(15 81 127 40)(16 21 128 82)(17 83 129 22)(18 23 130 84)(19 85 131 24)(20 25 132 86)(41 156 110 78)(42 79 111 157)(43 158 112 80)(44 61 113 159)(45 160 114 62)(46 63 115 141)(47 142 116 64)(48 65 117 143)(49 144 118 66)(50 67 119 145)(51 146 120 68)(52 69 101 147)(53 148 102 70)(54 71 103 149)(55 150 104 72)(56 73 105 151)(57 152 106 74)(58 75 107 153)(59 154 108 76)(60 77 109 155)
(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 10 123 132)(2 131 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 20 133 122)(12 121 134 19)(13 18 135 140)(14 139 136 17)(15 16 137 138)(21 81 92 30)(22 29 93 100)(23 99 94 28)(24 27 95 98)(25 97 96 26)(31 91 82 40)(32 39 83 90)(33 89 84 38)(34 37 85 88)(35 87 86 36)(41 105 120 46)(42 45 101 104)(43 103 102 44)(47 119 106 60)(48 59 107 118)(49 117 108 58)(50 57 109 116)(51 115 110 56)(52 55 111 114)(53 113 112 54)(61 70 149 158)(62 157 150 69)(63 68 151 156)(64 155 152 67)(65 66 153 154)(71 80 159 148)(72 147 160 79)(73 78 141 146)(74 145 142 77)(75 76 143 144)
G:=sub<Sym(160)| (1,149,123,61)(2,62,124,150)(3,151,125,63)(4,64,126,152)(5,153,127,65)(6,66,128,154)(7,155,129,67)(8,68,130,156)(9,157,131,69)(10,70,132,158)(11,159,133,71)(12,72,134,160)(13,141,135,73)(14,74,136,142)(15,143,137,75)(16,76,138,144)(17,145,139,77)(18,78,140,146)(19,147,121,79)(20,80,122,148)(21,59,92,118)(22,119,93,60)(23,41,94,120)(24,101,95,42)(25,43,96,102)(26,103,97,44)(27,45,98,104)(28,105,99,46)(29,47,100,106)(30,107,81,48)(31,49,82,108)(32,109,83,50)(33,51,84,110)(34,111,85,52)(35,53,86,112)(36,113,87,54)(37,55,88,114)(38,115,89,56)(39,57,90,116)(40,117,91,58), (1,87,133,26)(2,27,134,88)(3,89,135,28)(4,29,136,90)(5,91,137,30)(6,31,138,92)(7,93,139,32)(8,33,140,94)(9,95,121,34)(10,35,122,96)(11,97,123,36)(12,37,124,98)(13,99,125,38)(14,39,126,100)(15,81,127,40)(16,21,128,82)(17,83,129,22)(18,23,130,84)(19,85,131,24)(20,25,132,86)(41,156,110,78)(42,79,111,157)(43,158,112,80)(44,61,113,159)(45,160,114,62)(46,63,115,141)(47,142,116,64)(48,65,117,143)(49,144,118,66)(50,67,119,145)(51,146,120,68)(52,69,101,147)(53,148,102,70)(54,71,103,149)(55,150,104,72)(56,73,105,151)(57,152,106,74)(58,75,107,153)(59,154,108,76)(60,77,109,155), (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), (1,10,123,132)(2,131,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,20,133,122)(12,121,134,19)(13,18,135,140)(14,139,136,17)(15,16,137,138)(21,81,92,30)(22,29,93,100)(23,99,94,28)(24,27,95,98)(25,97,96,26)(31,91,82,40)(32,39,83,90)(33,89,84,38)(34,37,85,88)(35,87,86,36)(41,105,120,46)(42,45,101,104)(43,103,102,44)(47,119,106,60)(48,59,107,118)(49,117,108,58)(50,57,109,116)(51,115,110,56)(52,55,111,114)(53,113,112,54)(61,70,149,158)(62,157,150,69)(63,68,151,156)(64,155,152,67)(65,66,153,154)(71,80,159,148)(72,147,160,79)(73,78,141,146)(74,145,142,77)(75,76,143,144)>;
G:=Group( (1,149,123,61)(2,62,124,150)(3,151,125,63)(4,64,126,152)(5,153,127,65)(6,66,128,154)(7,155,129,67)(8,68,130,156)(9,157,131,69)(10,70,132,158)(11,159,133,71)(12,72,134,160)(13,141,135,73)(14,74,136,142)(15,143,137,75)(16,76,138,144)(17,145,139,77)(18,78,140,146)(19,147,121,79)(20,80,122,148)(21,59,92,118)(22,119,93,60)(23,41,94,120)(24,101,95,42)(25,43,96,102)(26,103,97,44)(27,45,98,104)(28,105,99,46)(29,47,100,106)(30,107,81,48)(31,49,82,108)(32,109,83,50)(33,51,84,110)(34,111,85,52)(35,53,86,112)(36,113,87,54)(37,55,88,114)(38,115,89,56)(39,57,90,116)(40,117,91,58), (1,87,133,26)(2,27,134,88)(3,89,135,28)(4,29,136,90)(5,91,137,30)(6,31,138,92)(7,93,139,32)(8,33,140,94)(9,95,121,34)(10,35,122,96)(11,97,123,36)(12,37,124,98)(13,99,125,38)(14,39,126,100)(15,81,127,40)(16,21,128,82)(17,83,129,22)(18,23,130,84)(19,85,131,24)(20,25,132,86)(41,156,110,78)(42,79,111,157)(43,158,112,80)(44,61,113,159)(45,160,114,62)(46,63,115,141)(47,142,116,64)(48,65,117,143)(49,144,118,66)(50,67,119,145)(51,146,120,68)(52,69,101,147)(53,148,102,70)(54,71,103,149)(55,150,104,72)(56,73,105,151)(57,152,106,74)(58,75,107,153)(59,154,108,76)(60,77,109,155), (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), (1,10,123,132)(2,131,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,20,133,122)(12,121,134,19)(13,18,135,140)(14,139,136,17)(15,16,137,138)(21,81,92,30)(22,29,93,100)(23,99,94,28)(24,27,95,98)(25,97,96,26)(31,91,82,40)(32,39,83,90)(33,89,84,38)(34,37,85,88)(35,87,86,36)(41,105,120,46)(42,45,101,104)(43,103,102,44)(47,119,106,60)(48,59,107,118)(49,117,108,58)(50,57,109,116)(51,115,110,56)(52,55,111,114)(53,113,112,54)(61,70,149,158)(62,157,150,69)(63,68,151,156)(64,155,152,67)(65,66,153,154)(71,80,159,148)(72,147,160,79)(73,78,141,146)(74,145,142,77)(75,76,143,144) );
G=PermutationGroup([[(1,149,123,61),(2,62,124,150),(3,151,125,63),(4,64,126,152),(5,153,127,65),(6,66,128,154),(7,155,129,67),(8,68,130,156),(9,157,131,69),(10,70,132,158),(11,159,133,71),(12,72,134,160),(13,141,135,73),(14,74,136,142),(15,143,137,75),(16,76,138,144),(17,145,139,77),(18,78,140,146),(19,147,121,79),(20,80,122,148),(21,59,92,118),(22,119,93,60),(23,41,94,120),(24,101,95,42),(25,43,96,102),(26,103,97,44),(27,45,98,104),(28,105,99,46),(29,47,100,106),(30,107,81,48),(31,49,82,108),(32,109,83,50),(33,51,84,110),(34,111,85,52),(35,53,86,112),(36,113,87,54),(37,55,88,114),(38,115,89,56),(39,57,90,116),(40,117,91,58)], [(1,87,133,26),(2,27,134,88),(3,89,135,28),(4,29,136,90),(5,91,137,30),(6,31,138,92),(7,93,139,32),(8,33,140,94),(9,95,121,34),(10,35,122,96),(11,97,123,36),(12,37,124,98),(13,99,125,38),(14,39,126,100),(15,81,127,40),(16,21,128,82),(17,83,129,22),(18,23,130,84),(19,85,131,24),(20,25,132,86),(41,156,110,78),(42,79,111,157),(43,158,112,80),(44,61,113,159),(45,160,114,62),(46,63,115,141),(47,142,116,64),(48,65,117,143),(49,144,118,66),(50,67,119,145),(51,146,120,68),(52,69,101,147),(53,148,102,70),(54,71,103,149),(55,150,104,72),(56,73,105,151),(57,152,106,74),(58,75,107,153),(59,154,108,76),(60,77,109,155)], [(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)], [(1,10,123,132),(2,131,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,20,133,122),(12,121,134,19),(13,18,135,140),(14,139,136,17),(15,16,137,138),(21,81,92,30),(22,29,93,100),(23,99,94,28),(24,27,95,98),(25,97,96,26),(31,91,82,40),(32,39,83,90),(33,89,84,38),(34,37,85,88),(35,87,86,36),(41,105,120,46),(42,45,101,104),(43,103,102,44),(47,119,106,60),(48,59,107,118),(49,117,108,58),(50,57,109,116),(51,115,110,56),(52,55,111,114),(53,113,112,54),(61,70,149,158),(62,157,150,69),(63,68,151,156),(64,155,152,67),(65,66,153,154),(71,80,159,148),(72,147,160,79),(73,78,141,146),(74,145,142,77),(75,76,143,144)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 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 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- 1+4 | D4⋊2D5 | Q8⋊2D5 | Q8.10D10 |
| kernel | C42.177D10 | C4×Dic10 | C4×D20 | C4⋊C4⋊7D5 | D10⋊2Q8 | C4⋊C4⋊D5 | Q8×Dic5 | C20.23D4 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 2 | 8 | 2 | 8 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of C42.177D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 6 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 35 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 9 | 0 | 0 |
| 0 | 0 | 1 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 26 | 37 | 0 | 0 | 0 | 0 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 40 | 0 | 0 |
| 0 | 0 | 11 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 15 | 4 | 0 | 0 | 0 | 0 |
| 5 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 32 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
G:=sub<GL(6,GF(41))| [32,6,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,35,0,0,0,0,0,32,0,0,0,0,0,0,21,1,0,0,0,0,9,20,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[26,15,0,0,0,0,37,15,0,0,0,0,0,0,25,11,0,0,0,0,40,16,0,0,0,0,0,0,40,34,0,0,0,0,7,7],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,16,32,0,0,0,0,1,25,0,0,0,0,0,0,40,34,0,0,0,0,0,1] >;
C42.177D10 in GAP, Magma, Sage, TeX
C_4^2._{177}D_{10} % in TeX
G:=Group("C4^2.177D10"); // GroupNames label
G:=SmallGroup(320,1404);
// by ID
G=gap.SmallGroup(320,1404);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations