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×C10).147C23, (C22×C20).234C22, 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, C10.33(C2×C4○D4), C2.36(C2×C4○D20), (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
Subgroups: 662 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, D5, C10 [×3], C10, C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×5], D10 [×3], C2×C10, C2×C10 [×3], C42⋊C2, C42⋊C2, C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×6], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×C10, C22.50C24, C4×Dic5, C4×Dic5 [×4], C10.D4, C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C4×Dic10 [×2], C42⋊2D5 [×2], C23.D10 [×2], Dic5.5D4 [×2], Dic5⋊3Q8, C20⋊Q8, C4⋊C4⋊7D5, D10⋊2Q8, C20.48D4, C4×C5⋊D4, C5×C42⋊C2, C42.98D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D5 [×7], C22.50C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.98D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 59 144 80)(2 50 145 71)(3 41 146 62)(4 52 147 73)(5 43 148 64)(6 54 149 75)(7 45 150 66)(8 56 151 77)(9 47 152 68)(10 58 153 79)(11 49 154 70)(12 60 155 61)(13 51 156 72)(14 42 157 63)(15 53 158 74)(16 44 159 65)(17 55 160 76)(18 46 141 67)(19 57 142 78)(20 48 143 69)(21 127 109 100)(22 138 110 91)(23 129 111 82)(24 140 112 93)(25 131 113 84)(26 122 114 95)(27 133 115 86)(28 124 116 97)(29 135 117 88)(30 126 118 99)(31 137 119 90)(32 128 120 81)(33 139 101 92)(34 130 102 83)(35 121 103 94)(36 132 104 85)(37 123 105 96)(38 134 106 87)(39 125 107 98)(40 136 108 89)
(1 139 11 129)(2 140 12 130)(3 121 13 131)(4 122 14 132)(5 123 15 133)(6 124 16 134)(7 125 17 135)(8 126 18 136)(9 127 19 137)(10 128 20 138)(21 78 31 68)(22 79 32 69)(23 80 33 70)(24 61 34 71)(25 62 35 72)(26 63 36 73)(27 64 37 74)(28 65 38 75)(29 66 39 76)(30 67 40 77)(41 103 51 113)(42 104 52 114)(43 105 53 115)(44 106 54 116)(45 107 55 117)(46 108 56 118)(47 109 57 119)(48 110 58 120)(49 111 59 101)(50 112 60 102)(81 143 91 153)(82 144 92 154)(83 145 93 155)(84 146 94 156)(85 147 95 157)(86 148 96 158)(87 149 97 159)(88 150 98 160)(89 151 99 141)(90 152 100 142)
(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)(21 102 31 112)(22 111 32 101)(23 120 33 110)(24 109 34 119)(25 118 35 108)(26 107 36 117)(27 116 37 106)(28 105 38 115)(29 114 39 104)(30 103 40 113)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)(81 139 91 129)(82 128 92 138)(83 137 93 127)(84 126 94 136)(85 135 95 125)(86 124 96 134)(87 133 97 123)(88 122 98 132)(89 131 99 121)(90 140 100 130)(141 146 151 156)(142 155 152 145)(143 144 153 154)(147 160 157 150)(148 149 158 159)
G:=sub<Sym(160)| (1,59,144,80)(2,50,145,71)(3,41,146,62)(4,52,147,73)(5,43,148,64)(6,54,149,75)(7,45,150,66)(8,56,151,77)(9,47,152,68)(10,58,153,79)(11,49,154,70)(12,60,155,61)(13,51,156,72)(14,42,157,63)(15,53,158,74)(16,44,159,65)(17,55,160,76)(18,46,141,67)(19,57,142,78)(20,48,143,69)(21,127,109,100)(22,138,110,91)(23,129,111,82)(24,140,112,93)(25,131,113,84)(26,122,114,95)(27,133,115,86)(28,124,116,97)(29,135,117,88)(30,126,118,99)(31,137,119,90)(32,128,120,81)(33,139,101,92)(34,130,102,83)(35,121,103,94)(36,132,104,85)(37,123,105,96)(38,134,106,87)(39,125,107,98)(40,136,108,89), (1,139,11,129)(2,140,12,130)(3,121,13,131)(4,122,14,132)(5,123,15,133)(6,124,16,134)(7,125,17,135)(8,126,18,136)(9,127,19,137)(10,128,20,138)(21,78,31,68)(22,79,32,69)(23,80,33,70)(24,61,34,71)(25,62,35,72)(26,63,36,73)(27,64,37,74)(28,65,38,75)(29,66,39,76)(30,67,40,77)(41,103,51,113)(42,104,52,114)(43,105,53,115)(44,106,54,116)(45,107,55,117)(46,108,56,118)(47,109,57,119)(48,110,58,120)(49,111,59,101)(50,112,60,102)(81,143,91,153)(82,144,92,154)(83,145,93,155)(84,146,94,156)(85,147,95,157)(86,148,96,158)(87,149,97,159)(88,150,98,160)(89,151,99,141)(90,152,100,142), (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)(21,102,31,112)(22,111,32,101)(23,120,33,110)(24,109,34,119)(25,118,35,108)(26,107,36,117)(27,116,37,106)(28,105,38,115)(29,114,39,104)(30,103,40,113)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,139,91,129)(82,128,92,138)(83,137,93,127)(84,126,94,136)(85,135,95,125)(86,124,96,134)(87,133,97,123)(88,122,98,132)(89,131,99,121)(90,140,100,130)(141,146,151,156)(142,155,152,145)(143,144,153,154)(147,160,157,150)(148,149,158,159)>;
G:=Group( (1,59,144,80)(2,50,145,71)(3,41,146,62)(4,52,147,73)(5,43,148,64)(6,54,149,75)(7,45,150,66)(8,56,151,77)(9,47,152,68)(10,58,153,79)(11,49,154,70)(12,60,155,61)(13,51,156,72)(14,42,157,63)(15,53,158,74)(16,44,159,65)(17,55,160,76)(18,46,141,67)(19,57,142,78)(20,48,143,69)(21,127,109,100)(22,138,110,91)(23,129,111,82)(24,140,112,93)(25,131,113,84)(26,122,114,95)(27,133,115,86)(28,124,116,97)(29,135,117,88)(30,126,118,99)(31,137,119,90)(32,128,120,81)(33,139,101,92)(34,130,102,83)(35,121,103,94)(36,132,104,85)(37,123,105,96)(38,134,106,87)(39,125,107,98)(40,136,108,89), (1,139,11,129)(2,140,12,130)(3,121,13,131)(4,122,14,132)(5,123,15,133)(6,124,16,134)(7,125,17,135)(8,126,18,136)(9,127,19,137)(10,128,20,138)(21,78,31,68)(22,79,32,69)(23,80,33,70)(24,61,34,71)(25,62,35,72)(26,63,36,73)(27,64,37,74)(28,65,38,75)(29,66,39,76)(30,67,40,77)(41,103,51,113)(42,104,52,114)(43,105,53,115)(44,106,54,116)(45,107,55,117)(46,108,56,118)(47,109,57,119)(48,110,58,120)(49,111,59,101)(50,112,60,102)(81,143,91,153)(82,144,92,154)(83,145,93,155)(84,146,94,156)(85,147,95,157)(86,148,96,158)(87,149,97,159)(88,150,98,160)(89,151,99,141)(90,152,100,142), (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)(21,102,31,112)(22,111,32,101)(23,120,33,110)(24,109,34,119)(25,118,35,108)(26,107,36,117)(27,116,37,106)(28,105,38,115)(29,114,39,104)(30,103,40,113)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,139,91,129)(82,128,92,138)(83,137,93,127)(84,126,94,136)(85,135,95,125)(86,124,96,134)(87,133,97,123)(88,122,98,132)(89,131,99,121)(90,140,100,130)(141,146,151,156)(142,155,152,145)(143,144,153,154)(147,160,157,150)(148,149,158,159) );
G=PermutationGroup([(1,59,144,80),(2,50,145,71),(3,41,146,62),(4,52,147,73),(5,43,148,64),(6,54,149,75),(7,45,150,66),(8,56,151,77),(9,47,152,68),(10,58,153,79),(11,49,154,70),(12,60,155,61),(13,51,156,72),(14,42,157,63),(15,53,158,74),(16,44,159,65),(17,55,160,76),(18,46,141,67),(19,57,142,78),(20,48,143,69),(21,127,109,100),(22,138,110,91),(23,129,111,82),(24,140,112,93),(25,131,113,84),(26,122,114,95),(27,133,115,86),(28,124,116,97),(29,135,117,88),(30,126,118,99),(31,137,119,90),(32,128,120,81),(33,139,101,92),(34,130,102,83),(35,121,103,94),(36,132,104,85),(37,123,105,96),(38,134,106,87),(39,125,107,98),(40,136,108,89)], [(1,139,11,129),(2,140,12,130),(3,121,13,131),(4,122,14,132),(5,123,15,133),(6,124,16,134),(7,125,17,135),(8,126,18,136),(9,127,19,137),(10,128,20,138),(21,78,31,68),(22,79,32,69),(23,80,33,70),(24,61,34,71),(25,62,35,72),(26,63,36,73),(27,64,37,74),(28,65,38,75),(29,66,39,76),(30,67,40,77),(41,103,51,113),(42,104,52,114),(43,105,53,115),(44,106,54,116),(45,107,55,117),(46,108,56,118),(47,109,57,119),(48,110,58,120),(49,111,59,101),(50,112,60,102),(81,143,91,153),(82,144,92,154),(83,145,93,155),(84,146,94,156),(85,147,95,157),(86,148,96,158),(87,149,97,159),(88,150,98,160),(89,151,99,141),(90,152,100,142)], [(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),(21,102,31,112),(22,111,32,101),(23,120,33,110),(24,109,34,119),(25,118,35,108),(26,107,36,117),(27,116,37,106),(28,105,38,115),(29,114,39,104),(30,103,40,113),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70),(81,139,91,129),(82,128,92,138),(83,137,93,127),(84,126,94,136),(85,135,95,125),(86,124,96,134),(87,133,97,123),(88,122,98,132),(89,131,99,121),(90,140,100,130),(141,146,151,156),(142,155,152,145),(143,144,153,154),(147,160,157,150),(148,149,158,159)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀