metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.98D10, C10.532- 1+4, C20⋊Q8⋊13C2, C4⋊C4.312D10, C42⋊2D5⋊6C2, D10⋊2Q8⋊13C2, (C4×Dic10)⋊11C2, C4.98(C4○D20), C42⋊C2⋊17D5, (C4×C20).28C22, (C2×C10).77C24, Dic5⋊3Q8⋊13C2, C20.200(C4○D4), C20.48D4⋊30C2, (C2×C20).698C23, C22⋊C4.101D10, (C22×C4).198D10, C23.D10⋊4C2, C23.88(C22×D5), Dic5.60(C4○D4), C4⋊Dic5.293C22, Dic5.5D4.1C2, (C2×Dic5).30C23, C10.D4.3C22, (C22×D5).25C23, C22.106(C23×D5), C23.D5.99C22, D10⋊C4.84C22, (C22×C20).234C22, (C22×C10).147C23, C5⋊1(C22.50C24), (C4×Dic5).219C22, C2.11(D4.10D10), (C2×Dic10).240C22, C2.16(D5×C4○D4), (C4×C5⋊D4).6C2, C4⋊C4⋊7D5⋊13C2, C2.36(C2×C4○D20), C10.33(C2×C4○D4), (C2×C4×D5).71C22, (C5×C42⋊C2)⋊19C2, (C5×C4⋊C4).313C22, (C2×C4).279(C22×D5), (C2×C5⋊D4).114C22, (C5×C22⋊C4).116C22, SmallGroup(320,1205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.98D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 662 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C22.50C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×Dic10, C42⋊2D5, C23.D10, Dic5.5D4, Dic5⋊3Q8, C20⋊Q8, C4⋊C4⋊7D5, D10⋊2Q8, C20.48D4, C4×C5⋊D4, C5×C42⋊C2, C42.98D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, C4○D20, C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.98D10
►On 160 points
Generators in S160
(1 99 22 149)(2 90 23 160)(3 81 24 151)(4 92 25 142)(5 83 26 153)(6 94 27 144)(7 85 28 155)(8 96 29 146)(9 87 30 157)(10 98 31 148)(11 89 32 159)(12 100 33 150)(13 91 34 141)(14 82 35 152)(15 93 36 143)(16 84 37 154)(17 95 38 145)(18 86 39 156)(19 97 40 147)(20 88 21 158)(41 112 122 66)(42 103 123 77)(43 114 124 68)(44 105 125 79)(45 116 126 70)(46 107 127 61)(47 118 128 72)(48 109 129 63)(49 120 130 74)(50 111 131 65)(51 102 132 76)(52 113 133 67)(53 104 134 78)(54 115 135 69)(55 106 136 80)(56 117 137 71)(57 108 138 62)(58 119 139 73)(59 110 140 64)(60 101 121 75)
(1 55 11 45)(2 56 12 46)(3 57 13 47)(4 58 14 48)(5 59 15 49)(6 60 16 50)(7 41 17 51)(8 42 18 52)(9 43 19 53)(10 44 20 54)(21 135 31 125)(22 136 32 126)(23 137 33 127)(24 138 34 128)(25 139 35 129)(26 140 36 130)(27 121 37 131)(28 122 38 132)(29 123 39 133)(30 124 40 134)(61 160 71 150)(62 141 72 151)(63 142 73 152)(64 143 74 153)(65 144 75 154)(66 145 76 155)(67 146 77 156)(68 147 78 157)(69 148 79 158)(70 149 80 159)(81 108 91 118)(82 109 92 119)(83 110 93 120)(84 111 94 101)(85 112 95 102)(86 113 96 103)(87 114 97 104)(88 115 98 105)(89 116 99 106)(90 117 100 107)
(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 22 31 32)(23 40 33 30)(24 29 34 39)(25 38 35 28)(26 27 36 37)(41 129 51 139)(42 138 52 128)(43 127 53 137)(44 136 54 126)(45 125 55 135)(46 134 56 124)(47 123 57 133)(48 132 58 122)(49 121 59 131)(50 130 60 140)(61 104 71 114)(62 113 72 103)(63 102 73 112)(64 111 74 101)(65 120 75 110)(66 109 76 119)(67 118 77 108)(68 107 78 117)(69 116 79 106)(70 105 80 115)(81 96 91 86)(82 85 92 95)(83 94 93 84)(87 90 97 100)(88 99 98 89)(141 156 151 146)(142 145 152 155)(143 154 153 144)(147 150 157 160)(148 159 158 149)
G:=sub<Sym(160)| (1,99,22,149)(2,90,23,160)(3,81,24,151)(4,92,25,142)(5,83,26,153)(6,94,27,144)(7,85,28,155)(8,96,29,146)(9,87,30,157)(10,98,31,148)(11,89,32,159)(12,100,33,150)(13,91,34,141)(14,82,35,152)(15,93,36,143)(16,84,37,154)(17,95,38,145)(18,86,39,156)(19,97,40,147)(20,88,21,158)(41,112,122,66)(42,103,123,77)(43,114,124,68)(44,105,125,79)(45,116,126,70)(46,107,127,61)(47,118,128,72)(48,109,129,63)(49,120,130,74)(50,111,131,65)(51,102,132,76)(52,113,133,67)(53,104,134,78)(54,115,135,69)(55,106,136,80)(56,117,137,71)(57,108,138,62)(58,119,139,73)(59,110,140,64)(60,101,121,75), (1,55,11,45)(2,56,12,46)(3,57,13,47)(4,58,14,48)(5,59,15,49)(6,60,16,50)(7,41,17,51)(8,42,18,52)(9,43,19,53)(10,44,20,54)(21,135,31,125)(22,136,32,126)(23,137,33,127)(24,138,34,128)(25,139,35,129)(26,140,36,130)(27,121,37,131)(28,122,38,132)(29,123,39,133)(30,124,40,134)(61,160,71,150)(62,141,72,151)(63,142,73,152)(64,143,74,153)(65,144,75,154)(66,145,76,155)(67,146,77,156)(68,147,78,157)(69,148,79,158)(70,149,80,159)(81,108,91,118)(82,109,92,119)(83,110,93,120)(84,111,94,101)(85,112,95,102)(86,113,96,103)(87,114,97,104)(88,115,98,105)(89,116,99,106)(90,117,100,107), (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)(21,22,31,32)(23,40,33,30)(24,29,34,39)(25,38,35,28)(26,27,36,37)(41,129,51,139)(42,138,52,128)(43,127,53,137)(44,136,54,126)(45,125,55,135)(46,134,56,124)(47,123,57,133)(48,132,58,122)(49,121,59,131)(50,130,60,140)(61,104,71,114)(62,113,72,103)(63,102,73,112)(64,111,74,101)(65,120,75,110)(66,109,76,119)(67,118,77,108)(68,107,78,117)(69,116,79,106)(70,105,80,115)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(141,156,151,146)(142,145,152,155)(143,154,153,144)(147,150,157,160)(148,159,158,149)>;
G:=Group( (1,99,22,149)(2,90,23,160)(3,81,24,151)(4,92,25,142)(5,83,26,153)(6,94,27,144)(7,85,28,155)(8,96,29,146)(9,87,30,157)(10,98,31,148)(11,89,32,159)(12,100,33,150)(13,91,34,141)(14,82,35,152)(15,93,36,143)(16,84,37,154)(17,95,38,145)(18,86,39,156)(19,97,40,147)(20,88,21,158)(41,112,122,66)(42,103,123,77)(43,114,124,68)(44,105,125,79)(45,116,126,70)(46,107,127,61)(47,118,128,72)(48,109,129,63)(49,120,130,74)(50,111,131,65)(51,102,132,76)(52,113,133,67)(53,104,134,78)(54,115,135,69)(55,106,136,80)(56,117,137,71)(57,108,138,62)(58,119,139,73)(59,110,140,64)(60,101,121,75), (1,55,11,45)(2,56,12,46)(3,57,13,47)(4,58,14,48)(5,59,15,49)(6,60,16,50)(7,41,17,51)(8,42,18,52)(9,43,19,53)(10,44,20,54)(21,135,31,125)(22,136,32,126)(23,137,33,127)(24,138,34,128)(25,139,35,129)(26,140,36,130)(27,121,37,131)(28,122,38,132)(29,123,39,133)(30,124,40,134)(61,160,71,150)(62,141,72,151)(63,142,73,152)(64,143,74,153)(65,144,75,154)(66,145,76,155)(67,146,77,156)(68,147,78,157)(69,148,79,158)(70,149,80,159)(81,108,91,118)(82,109,92,119)(83,110,93,120)(84,111,94,101)(85,112,95,102)(86,113,96,103)(87,114,97,104)(88,115,98,105)(89,116,99,106)(90,117,100,107), (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)(21,22,31,32)(23,40,33,30)(24,29,34,39)(25,38,35,28)(26,27,36,37)(41,129,51,139)(42,138,52,128)(43,127,53,137)(44,136,54,126)(45,125,55,135)(46,134,56,124)(47,123,57,133)(48,132,58,122)(49,121,59,131)(50,130,60,140)(61,104,71,114)(62,113,72,103)(63,102,73,112)(64,111,74,101)(65,120,75,110)(66,109,76,119)(67,118,77,108)(68,107,78,117)(69,116,79,106)(70,105,80,115)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(141,156,151,146)(142,145,152,155)(143,154,153,144)(147,150,157,160)(148,159,158,149) );
G=PermutationGroup([[(1,99,22,149),(2,90,23,160),(3,81,24,151),(4,92,25,142),(5,83,26,153),(6,94,27,144),(7,85,28,155),(8,96,29,146),(9,87,30,157),(10,98,31,148),(11,89,32,159),(12,100,33,150),(13,91,34,141),(14,82,35,152),(15,93,36,143),(16,84,37,154),(17,95,38,145),(18,86,39,156),(19,97,40,147),(20,88,21,158),(41,112,122,66),(42,103,123,77),(43,114,124,68),(44,105,125,79),(45,116,126,70),(46,107,127,61),(47,118,128,72),(48,109,129,63),(49,120,130,74),(50,111,131,65),(51,102,132,76),(52,113,133,67),(53,104,134,78),(54,115,135,69),(55,106,136,80),(56,117,137,71),(57,108,138,62),(58,119,139,73),(59,110,140,64),(60,101,121,75)], [(1,55,11,45),(2,56,12,46),(3,57,13,47),(4,58,14,48),(5,59,15,49),(6,60,16,50),(7,41,17,51),(8,42,18,52),(9,43,19,53),(10,44,20,54),(21,135,31,125),(22,136,32,126),(23,137,33,127),(24,138,34,128),(25,139,35,129),(26,140,36,130),(27,121,37,131),(28,122,38,132),(29,123,39,133),(30,124,40,134),(61,160,71,150),(62,141,72,151),(63,142,73,152),(64,143,74,153),(65,144,75,154),(66,145,76,155),(67,146,77,156),(68,147,78,157),(69,148,79,158),(70,149,80,159),(81,108,91,118),(82,109,92,119),(83,110,93,120),(84,111,94,101),(85,112,95,102),(86,113,96,103),(87,114,97,104),(88,115,98,105),(89,116,99,106),(90,117,100,107)], [(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),(21,22,31,32),(23,40,33,30),(24,29,34,39),(25,38,35,28),(26,27,36,37),(41,129,51,139),(42,138,52,128),(43,127,53,137),(44,136,54,126),(45,125,55,135),(46,134,56,124),(47,123,57,133),(48,132,58,122),(49,121,59,131),(50,130,60,140),(61,104,71,114),(62,113,72,103),(63,102,73,112),(64,111,74,101),(65,120,75,110),(66,109,76,119),(67,118,77,108),(68,107,78,117),(69,116,79,106),(70,105,80,115),(81,96,91,86),(82,85,92,95),(83,94,93,84),(87,90,97,100),(88,99,98,89),(141,156,151,146),(142,145,152,155),(143,154,153,144),(147,150,157,160),(148,159,158,149)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | ··· | 4S | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | D5×C4○D4 | D4.10D10 |
| kernel | C42.98D10 | C4×Dic10 | C42⋊2D5 | C23.D10 | Dic5.5D4 | Dic5⋊3Q8 | C20⋊Q8 | C4⋊C4⋊7D5 | D10⋊2Q8 | C20.48D4 | C4×C5⋊D4 | C5×C42⋊C2 | C42⋊C2 | Dic5 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.98D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 32 | 32 |
| 24 | 40 | 0 | 0 |
| 1 | 17 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 34 | 0 | 0 |
| 7 | 34 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 32 |
| 34 | 1 | 0 | 0 |
| 34 | 7 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,32,0,0,0,32],[24,1,0,0,40,17,0,0,0,0,9,0,0,0,0,9],[1,7,0,0,34,34,0,0,0,0,9,0,0,0,18,32],[34,34,0,0,1,7,0,0,0,0,9,0,0,0,0,9] >;
C42.98D10 in GAP, Magma, Sage, TeX
C_4^2._{98}D_{10} % in TeX
G:=Group("C4^2.98D10"); // GroupNames label
G:=SmallGroup(320,1205);
// by ID
G=gap.SmallGroup(320,1205);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,1571,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations