metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.153D10, C10.1332+ (1+4), (C4×D20)⋊48C2, C42.C2⋊9D5, C4⋊2D20⋊33C2, C4⋊C4.209D10, D20⋊8C4⋊37C2, (C2×C20).91C23, D10.55(C4○D4), C20.130(C4○D4), (C2×C10).239C24, (C4×C20).198C22, C4.39(Q8⋊2D5), D10.13D4⋊35C2, C2.58(D4⋊8D10), (C2×D20).276C22, C4⋊Dic5.315C22, C22.260(C23×D5), D10⋊C4.41C22, (C4×Dic5).153C22, (C2×Dic5).124C23, C10.D4.54C22, (C22×D5).104C23, C5⋊10(C22.47C24), (D5×C4⋊C4)⋊39C2, C2.90(D5×C4○D4), C4⋊C4⋊D5⋊37C2, C4⋊C4⋊7D5⋊38C2, C10.201(C2×C4○D4), C2.24(C2×Q8⋊2D5), (C5×C42.C2)⋊12C2, (C2×C4×D5).138C22, (C2×C4).82(C22×D5), (C5×C4⋊C4).194C22, SmallGroup(320,1367)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 950 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×12], D4 [×10], C23 [×4], D5 [×5], C10 [×3], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×6], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×11], C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5 [×8], D20 [×10], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], C22×D5 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×4], C4×D20 [×2], D5×C4⋊C4, C4⋊C4⋊7D5, D20⋊8C4 [×2], D10.13D4 [×2], C4⋊2D20 [×2], C4⋊2D20 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.153D10
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], C22.47C24, Q8⋊2D5 [×2], C23×D5, C2×Q8⋊2D5, D5×C4○D4, D4⋊8D10, C42.153D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 148 114 129)(2 130 115 149)(3 150 116 131)(4 132 117 151)(5 152 118 133)(6 134 119 153)(7 154 120 135)(8 136 101 155)(9 156 102 137)(10 138 103 157)(11 158 104 139)(12 140 105 159)(13 160 106 121)(14 122 107 141)(15 142 108 123)(16 124 109 143)(17 144 110 125)(18 126 111 145)(19 146 112 127)(20 128 113 147)(21 81 57 72)(22 73 58 82)(23 83 59 74)(24 75 60 84)(25 85 41 76)(26 77 42 86)(27 87 43 78)(28 79 44 88)(29 89 45 80)(30 61 46 90)(31 91 47 62)(32 63 48 92)(33 93 49 64)(34 65 50 94)(35 95 51 66)(36 67 52 96)(37 97 53 68)(38 69 54 98)(39 99 55 70)(40 71 56 100)
(1 58 104 32)(2 49 105 23)(3 60 106 34)(4 51 107 25)(5 42 108 36)(6 53 109 27)(7 44 110 38)(8 55 111 29)(9 46 112 40)(10 57 113 31)(11 48 114 22)(12 59 115 33)(13 50 116 24)(14 41 117 35)(15 52 118 26)(16 43 119 37)(17 54 120 28)(18 45 101 39)(19 56 102 30)(20 47 103 21)(61 146 100 137)(62 157 81 128)(63 148 82 139)(64 159 83 130)(65 150 84 121)(66 141 85 132)(67 152 86 123)(68 143 87 134)(69 154 88 125)(70 145 89 136)(71 156 90 127)(72 147 91 138)(73 158 92 129)(74 149 93 140)(75 160 94 131)(76 151 95 122)(77 142 96 133)(78 153 97 124)(79 144 98 135)(80 155 99 126)
(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 67 11 77)(2 76 12 66)(3 65 13 75)(4 74 14 64)(5 63 15 73)(6 72 16 62)(7 61 17 71)(8 70 18 80)(9 79 19 69)(10 68 20 78)(21 143 31 153)(22 152 32 142)(23 141 33 151)(24 150 34 160)(25 159 35 149)(26 148 36 158)(27 157 37 147)(28 146 38 156)(29 155 39 145)(30 144 40 154)(41 140 51 130)(42 129 52 139)(43 138 53 128)(44 127 54 137)(45 136 55 126)(46 125 56 135)(47 134 57 124)(48 123 58 133)(49 132 59 122)(50 121 60 131)(81 109 91 119)(82 118 92 108)(83 107 93 117)(84 116 94 106)(85 105 95 115)(86 114 96 104)(87 103 97 113)(88 112 98 102)(89 101 99 111)(90 110 100 120)
G:=sub<Sym(160)| (1,148,114,129)(2,130,115,149)(3,150,116,131)(4,132,117,151)(5,152,118,133)(6,134,119,153)(7,154,120,135)(8,136,101,155)(9,156,102,137)(10,138,103,157)(11,158,104,139)(12,140,105,159)(13,160,106,121)(14,122,107,141)(15,142,108,123)(16,124,109,143)(17,144,110,125)(18,126,111,145)(19,146,112,127)(20,128,113,147)(21,81,57,72)(22,73,58,82)(23,83,59,74)(24,75,60,84)(25,85,41,76)(26,77,42,86)(27,87,43,78)(28,79,44,88)(29,89,45,80)(30,61,46,90)(31,91,47,62)(32,63,48,92)(33,93,49,64)(34,65,50,94)(35,95,51,66)(36,67,52,96)(37,97,53,68)(38,69,54,98)(39,99,55,70)(40,71,56,100), (1,58,104,32)(2,49,105,23)(3,60,106,34)(4,51,107,25)(5,42,108,36)(6,53,109,27)(7,44,110,38)(8,55,111,29)(9,46,112,40)(10,57,113,31)(11,48,114,22)(12,59,115,33)(13,50,116,24)(14,41,117,35)(15,52,118,26)(16,43,119,37)(17,54,120,28)(18,45,101,39)(19,56,102,30)(20,47,103,21)(61,146,100,137)(62,157,81,128)(63,148,82,139)(64,159,83,130)(65,150,84,121)(66,141,85,132)(67,152,86,123)(68,143,87,134)(69,154,88,125)(70,145,89,136)(71,156,90,127)(72,147,91,138)(73,158,92,129)(74,149,93,140)(75,160,94,131)(76,151,95,122)(77,142,96,133)(78,153,97,124)(79,144,98,135)(80,155,99,126), (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), (1,67,11,77)(2,76,12,66)(3,65,13,75)(4,74,14,64)(5,63,15,73)(6,72,16,62)(7,61,17,71)(8,70,18,80)(9,79,19,69)(10,68,20,78)(21,143,31,153)(22,152,32,142)(23,141,33,151)(24,150,34,160)(25,159,35,149)(26,148,36,158)(27,157,37,147)(28,146,38,156)(29,155,39,145)(30,144,40,154)(41,140,51,130)(42,129,52,139)(43,138,53,128)(44,127,54,137)(45,136,55,126)(46,125,56,135)(47,134,57,124)(48,123,58,133)(49,132,59,122)(50,121,60,131)(81,109,91,119)(82,118,92,108)(83,107,93,117)(84,116,94,106)(85,105,95,115)(86,114,96,104)(87,103,97,113)(88,112,98,102)(89,101,99,111)(90,110,100,120)>;
G:=Group( (1,148,114,129)(2,130,115,149)(3,150,116,131)(4,132,117,151)(5,152,118,133)(6,134,119,153)(7,154,120,135)(8,136,101,155)(9,156,102,137)(10,138,103,157)(11,158,104,139)(12,140,105,159)(13,160,106,121)(14,122,107,141)(15,142,108,123)(16,124,109,143)(17,144,110,125)(18,126,111,145)(19,146,112,127)(20,128,113,147)(21,81,57,72)(22,73,58,82)(23,83,59,74)(24,75,60,84)(25,85,41,76)(26,77,42,86)(27,87,43,78)(28,79,44,88)(29,89,45,80)(30,61,46,90)(31,91,47,62)(32,63,48,92)(33,93,49,64)(34,65,50,94)(35,95,51,66)(36,67,52,96)(37,97,53,68)(38,69,54,98)(39,99,55,70)(40,71,56,100), (1,58,104,32)(2,49,105,23)(3,60,106,34)(4,51,107,25)(5,42,108,36)(6,53,109,27)(7,44,110,38)(8,55,111,29)(9,46,112,40)(10,57,113,31)(11,48,114,22)(12,59,115,33)(13,50,116,24)(14,41,117,35)(15,52,118,26)(16,43,119,37)(17,54,120,28)(18,45,101,39)(19,56,102,30)(20,47,103,21)(61,146,100,137)(62,157,81,128)(63,148,82,139)(64,159,83,130)(65,150,84,121)(66,141,85,132)(67,152,86,123)(68,143,87,134)(69,154,88,125)(70,145,89,136)(71,156,90,127)(72,147,91,138)(73,158,92,129)(74,149,93,140)(75,160,94,131)(76,151,95,122)(77,142,96,133)(78,153,97,124)(79,144,98,135)(80,155,99,126), (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), (1,67,11,77)(2,76,12,66)(3,65,13,75)(4,74,14,64)(5,63,15,73)(6,72,16,62)(7,61,17,71)(8,70,18,80)(9,79,19,69)(10,68,20,78)(21,143,31,153)(22,152,32,142)(23,141,33,151)(24,150,34,160)(25,159,35,149)(26,148,36,158)(27,157,37,147)(28,146,38,156)(29,155,39,145)(30,144,40,154)(41,140,51,130)(42,129,52,139)(43,138,53,128)(44,127,54,137)(45,136,55,126)(46,125,56,135)(47,134,57,124)(48,123,58,133)(49,132,59,122)(50,121,60,131)(81,109,91,119)(82,118,92,108)(83,107,93,117)(84,116,94,106)(85,105,95,115)(86,114,96,104)(87,103,97,113)(88,112,98,102)(89,101,99,111)(90,110,100,120) );
G=PermutationGroup([(1,148,114,129),(2,130,115,149),(3,150,116,131),(4,132,117,151),(5,152,118,133),(6,134,119,153),(7,154,120,135),(8,136,101,155),(9,156,102,137),(10,138,103,157),(11,158,104,139),(12,140,105,159),(13,160,106,121),(14,122,107,141),(15,142,108,123),(16,124,109,143),(17,144,110,125),(18,126,111,145),(19,146,112,127),(20,128,113,147),(21,81,57,72),(22,73,58,82),(23,83,59,74),(24,75,60,84),(25,85,41,76),(26,77,42,86),(27,87,43,78),(28,79,44,88),(29,89,45,80),(30,61,46,90),(31,91,47,62),(32,63,48,92),(33,93,49,64),(34,65,50,94),(35,95,51,66),(36,67,52,96),(37,97,53,68),(38,69,54,98),(39,99,55,70),(40,71,56,100)], [(1,58,104,32),(2,49,105,23),(3,60,106,34),(4,51,107,25),(5,42,108,36),(6,53,109,27),(7,44,110,38),(8,55,111,29),(9,46,112,40),(10,57,113,31),(11,48,114,22),(12,59,115,33),(13,50,116,24),(14,41,117,35),(15,52,118,26),(16,43,119,37),(17,54,120,28),(18,45,101,39),(19,56,102,30),(20,47,103,21),(61,146,100,137),(62,157,81,128),(63,148,82,139),(64,159,83,130),(65,150,84,121),(66,141,85,132),(67,152,86,123),(68,143,87,134),(69,154,88,125),(70,145,89,136),(71,156,90,127),(72,147,91,138),(73,158,92,129),(74,149,93,140),(75,160,94,131),(76,151,95,122),(77,142,96,133),(78,153,97,124),(79,144,98,135),(80,155,99,126)], [(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)], [(1,67,11,77),(2,76,12,66),(3,65,13,75),(4,74,14,64),(5,63,15,73),(6,72,16,62),(7,61,17,71),(8,70,18,80),(9,79,19,69),(10,68,20,78),(21,143,31,153),(22,152,32,142),(23,141,33,151),(24,150,34,160),(25,159,35,149),(26,148,36,158),(27,157,37,147),(28,146,38,156),(29,155,39,145),(30,144,40,154),(41,140,51,130),(42,129,52,139),(43,138,53,128),(44,127,54,137),(45,136,55,126),(46,125,56,135),(47,134,57,124),(48,123,58,133),(49,132,59,122),(50,121,60,131),(81,109,91,119),(82,118,92,108),(83,107,93,117),(84,116,94,106),(85,105,95,115),(86,114,96,104),(87,103,97,113),(88,112,98,102),(89,101,99,111),(90,110,100,120)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 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 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 12 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 29 | 18 | 0 | 0 | 0 | 0 |
| 17 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0],[9,12,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[29,17,0,0,0,0,18,12,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 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 | C4○D4 | D10 | D10 | 2+ (1+4) | Q8⋊2D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.153D10 | C4×D20 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C20 | D10 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{153}D_{10} % in TeX
G:=Group("C4^2.153D10"); // GroupNames label
G:=SmallGroup(320,1367);
// by ID
G=gap.SmallGroup(320,1367);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀