metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.133D10, C10.132- (1+4), C10.1122+ (1+4), C20⋊Q8⋊17C2, (C4×Q8)⋊15D5, (C4×D20)⋊41C2, (Q8×C20)⋊17C2, C4⋊C4.300D10, D10⋊Q8⋊12C2, D10⋊3Q8⋊10C2, C4.49(C4○D20), C4.D20⋊29C2, C4⋊2D20.11C2, C20.23D4⋊9C2, C42⋊D5⋊18C2, C42⋊2D5⋊12C2, (C2×Q8).181D10, D10.13D4⋊9C2, C20.120(C4○D4), (C2×C20).171C23, (C4×C20).178C22, (C2×C10).126C24, C2.24(D4⋊8D10), (C2×D20).271C22, D10⋊C4.7C22, C4⋊Dic5.369C22, (Q8×C10).226C22, (C4×Dic5).94C22, (C2×Dic5).57C23, (C22×D5).48C23, C22.147(C23×D5), C5⋊3(C22.36C24), (C2×Dic10).34C22, C10.D4.77C22, C2.14(Q8.10D10), C10.56(C2×C4○D4), C2.65(C2×C4○D20), (C2×C4×D5).85C22, (C5×C4⋊C4).354C22, (C2×C4).171(C22×D5), SmallGroup(320,1254)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 814 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], D5 [×3], C10 [×3], C42, C42 [×2], C42, C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic5 [×5], C20 [×2], C20 [×6], D10 [×9], C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22.36C24, C4×Dic5, C10.D4 [×2], C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×2], D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q8×C10, C42⋊D5, C4×D20, C4.D20 [×2], C42⋊2D5 [×2], C20⋊Q8, D10.13D4 [×2], C4⋊2D20, D10⋊Q8 [×2], D10⋊3Q8, C20.23D4, Q8×C20, C42.133D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.36C24, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D4⋊8D10, C42.133D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 80 11 70)(2 71 12 61)(3 62 13 72)(4 73 14 63)(5 64 15 74)(6 75 16 65)(7 66 17 76)(8 77 18 67)(9 68 19 78)(10 79 20 69)(21 107 31 117)(22 118 32 108)(23 109 33 119)(24 120 34 110)(25 111 35 101)(26 102 36 112)(27 113 37 103)(28 104 38 114)(29 115 39 105)(30 106 40 116)(41 83 51 93)(42 94 52 84)(43 85 53 95)(44 96 54 86)(45 87 55 97)(46 98 56 88)(47 89 57 99)(48 100 58 90)(49 91 59 81)(50 82 60 92)(121 145 131 155)(122 156 132 146)(123 147 133 157)(124 158 134 148)(125 149 135 159)(126 160 136 150)(127 151 137 141)(128 142 138 152)(129 153 139 143)(130 144 140 154)
(1 49 141 37)(2 50 142 38)(3 51 143 39)(4 52 144 40)(5 53 145 21)(6 54 146 22)(7 55 147 23)(8 56 148 24)(9 57 149 25)(10 58 150 26)(11 59 151 27)(12 60 152 28)(13 41 153 29)(14 42 154 30)(15 43 155 31)(16 44 156 32)(17 45 157 33)(18 46 158 34)(19 47 159 35)(20 48 160 36)(61 92 128 104)(62 93 129 105)(63 94 130 106)(64 95 131 107)(65 96 132 108)(66 97 133 109)(67 98 134 110)(68 99 135 111)(69 100 136 112)(70 81 137 113)(71 82 138 114)(72 83 139 115)(73 84 140 116)(74 85 121 117)(75 86 122 118)(76 87 123 119)(77 88 124 120)(78 89 125 101)(79 90 126 102)(80 91 127 103)
(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)
(1 107 151 85)(2 84 152 106)(3 105 153 83)(4 82 154 104)(5 103 155 81)(6 100 156 102)(7 101 157 99)(8 98 158 120)(9 119 159 97)(10 96 160 118)(11 117 141 95)(12 94 142 116)(13 115 143 93)(14 92 144 114)(15 113 145 91)(16 90 146 112)(17 111 147 89)(18 88 148 110)(19 109 149 87)(20 86 150 108)(21 137 43 80)(22 79 44 136)(23 135 45 78)(24 77 46 134)(25 133 47 76)(26 75 48 132)(27 131 49 74)(28 73 50 130)(29 129 51 72)(30 71 52 128)(31 127 53 70)(32 69 54 126)(33 125 55 68)(34 67 56 124)(35 123 57 66)(36 65 58 122)(37 121 59 64)(38 63 60 140)(39 139 41 62)(40 61 42 138)
G:=sub<Sym(160)| (1,80,11,70)(2,71,12,61)(3,62,13,72)(4,73,14,63)(5,64,15,74)(6,75,16,65)(7,66,17,76)(8,77,18,67)(9,68,19,78)(10,79,20,69)(21,107,31,117)(22,118,32,108)(23,109,33,119)(24,120,34,110)(25,111,35,101)(26,102,36,112)(27,113,37,103)(28,104,38,114)(29,115,39,105)(30,106,40,116)(41,83,51,93)(42,94,52,84)(43,85,53,95)(44,96,54,86)(45,87,55,97)(46,98,56,88)(47,89,57,99)(48,100,58,90)(49,91,59,81)(50,82,60,92)(121,145,131,155)(122,156,132,146)(123,147,133,157)(124,158,134,148)(125,149,135,159)(126,160,136,150)(127,151,137,141)(128,142,138,152)(129,153,139,143)(130,144,140,154), (1,49,141,37)(2,50,142,38)(3,51,143,39)(4,52,144,40)(5,53,145,21)(6,54,146,22)(7,55,147,23)(8,56,148,24)(9,57,149,25)(10,58,150,26)(11,59,151,27)(12,60,152,28)(13,41,153,29)(14,42,154,30)(15,43,155,31)(16,44,156,32)(17,45,157,33)(18,46,158,34)(19,47,159,35)(20,48,160,36)(61,92,128,104)(62,93,129,105)(63,94,130,106)(64,95,131,107)(65,96,132,108)(66,97,133,109)(67,98,134,110)(68,99,135,111)(69,100,136,112)(70,81,137,113)(71,82,138,114)(72,83,139,115)(73,84,140,116)(74,85,121,117)(75,86,122,118)(76,87,123,119)(77,88,124,120)(78,89,125,101)(79,90,126,102)(80,91,127,103), (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), (1,107,151,85)(2,84,152,106)(3,105,153,83)(4,82,154,104)(5,103,155,81)(6,100,156,102)(7,101,157,99)(8,98,158,120)(9,119,159,97)(10,96,160,118)(11,117,141,95)(12,94,142,116)(13,115,143,93)(14,92,144,114)(15,113,145,91)(16,90,146,112)(17,111,147,89)(18,88,148,110)(19,109,149,87)(20,86,150,108)(21,137,43,80)(22,79,44,136)(23,135,45,78)(24,77,46,134)(25,133,47,76)(26,75,48,132)(27,131,49,74)(28,73,50,130)(29,129,51,72)(30,71,52,128)(31,127,53,70)(32,69,54,126)(33,125,55,68)(34,67,56,124)(35,123,57,66)(36,65,58,122)(37,121,59,64)(38,63,60,140)(39,139,41,62)(40,61,42,138)>;
G:=Group( (1,80,11,70)(2,71,12,61)(3,62,13,72)(4,73,14,63)(5,64,15,74)(6,75,16,65)(7,66,17,76)(8,77,18,67)(9,68,19,78)(10,79,20,69)(21,107,31,117)(22,118,32,108)(23,109,33,119)(24,120,34,110)(25,111,35,101)(26,102,36,112)(27,113,37,103)(28,104,38,114)(29,115,39,105)(30,106,40,116)(41,83,51,93)(42,94,52,84)(43,85,53,95)(44,96,54,86)(45,87,55,97)(46,98,56,88)(47,89,57,99)(48,100,58,90)(49,91,59,81)(50,82,60,92)(121,145,131,155)(122,156,132,146)(123,147,133,157)(124,158,134,148)(125,149,135,159)(126,160,136,150)(127,151,137,141)(128,142,138,152)(129,153,139,143)(130,144,140,154), (1,49,141,37)(2,50,142,38)(3,51,143,39)(4,52,144,40)(5,53,145,21)(6,54,146,22)(7,55,147,23)(8,56,148,24)(9,57,149,25)(10,58,150,26)(11,59,151,27)(12,60,152,28)(13,41,153,29)(14,42,154,30)(15,43,155,31)(16,44,156,32)(17,45,157,33)(18,46,158,34)(19,47,159,35)(20,48,160,36)(61,92,128,104)(62,93,129,105)(63,94,130,106)(64,95,131,107)(65,96,132,108)(66,97,133,109)(67,98,134,110)(68,99,135,111)(69,100,136,112)(70,81,137,113)(71,82,138,114)(72,83,139,115)(73,84,140,116)(74,85,121,117)(75,86,122,118)(76,87,123,119)(77,88,124,120)(78,89,125,101)(79,90,126,102)(80,91,127,103), (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), (1,107,151,85)(2,84,152,106)(3,105,153,83)(4,82,154,104)(5,103,155,81)(6,100,156,102)(7,101,157,99)(8,98,158,120)(9,119,159,97)(10,96,160,118)(11,117,141,95)(12,94,142,116)(13,115,143,93)(14,92,144,114)(15,113,145,91)(16,90,146,112)(17,111,147,89)(18,88,148,110)(19,109,149,87)(20,86,150,108)(21,137,43,80)(22,79,44,136)(23,135,45,78)(24,77,46,134)(25,133,47,76)(26,75,48,132)(27,131,49,74)(28,73,50,130)(29,129,51,72)(30,71,52,128)(31,127,53,70)(32,69,54,126)(33,125,55,68)(34,67,56,124)(35,123,57,66)(36,65,58,122)(37,121,59,64)(38,63,60,140)(39,139,41,62)(40,61,42,138) );
G=PermutationGroup([(1,80,11,70),(2,71,12,61),(3,62,13,72),(4,73,14,63),(5,64,15,74),(6,75,16,65),(7,66,17,76),(8,77,18,67),(9,68,19,78),(10,79,20,69),(21,107,31,117),(22,118,32,108),(23,109,33,119),(24,120,34,110),(25,111,35,101),(26,102,36,112),(27,113,37,103),(28,104,38,114),(29,115,39,105),(30,106,40,116),(41,83,51,93),(42,94,52,84),(43,85,53,95),(44,96,54,86),(45,87,55,97),(46,98,56,88),(47,89,57,99),(48,100,58,90),(49,91,59,81),(50,82,60,92),(121,145,131,155),(122,156,132,146),(123,147,133,157),(124,158,134,148),(125,149,135,159),(126,160,136,150),(127,151,137,141),(128,142,138,152),(129,153,139,143),(130,144,140,154)], [(1,49,141,37),(2,50,142,38),(3,51,143,39),(4,52,144,40),(5,53,145,21),(6,54,146,22),(7,55,147,23),(8,56,148,24),(9,57,149,25),(10,58,150,26),(11,59,151,27),(12,60,152,28),(13,41,153,29),(14,42,154,30),(15,43,155,31),(16,44,156,32),(17,45,157,33),(18,46,158,34),(19,47,159,35),(20,48,160,36),(61,92,128,104),(62,93,129,105),(63,94,130,106),(64,95,131,107),(65,96,132,108),(66,97,133,109),(67,98,134,110),(68,99,135,111),(69,100,136,112),(70,81,137,113),(71,82,138,114),(72,83,139,115),(73,84,140,116),(74,85,121,117),(75,86,122,118),(76,87,123,119),(77,88,124,120),(78,89,125,101),(79,90,126,102),(80,91,127,103)], [(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)], [(1,107,151,85),(2,84,152,106),(3,105,153,83),(4,82,154,104),(5,103,155,81),(6,100,156,102),(7,101,157,99),(8,98,158,120),(9,119,159,97),(10,96,160,118),(11,117,141,95),(12,94,142,116),(13,115,143,93),(14,92,144,114),(15,113,145,91),(16,90,146,112),(17,111,147,89),(18,88,148,110),(19,109,149,87),(20,86,150,108),(21,137,43,80),(22,79,44,136),(23,135,45,78),(24,77,46,134),(25,133,47,76),(26,75,48,132),(27,131,49,74),(28,73,50,130),(29,129,51,72),(30,71,52,128),(31,127,53,70),(32,69,54,126),(33,125,55,68),(34,67,56,124),(35,123,57,66),(36,65,58,122),(37,121,59,64),(38,63,60,140),(39,139,41,62),(40,61,42,138)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 14 | 31 | 14 |
| 0 | 0 | 10 | 3 | 10 | 4 |
| 0 | 0 | 33 | 15 | 11 | 27 |
| 0 | 0 | 40 | 7 | 31 | 38 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 32 | 0 | 0 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 28 | 0 | 39 | 32 |
| 0 | 0 | 28 | 13 | 37 | 2 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 9 | 37 | 13 |
| 0 | 0 | 4 | 4 | 0 | 9 |
| 0 | 0 | 2 | 15 | 5 | 32 |
| 0 | 0 | 40 | 17 | 20 | 13 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 4 | 5 | 36 |
| 0 | 0 | 3 | 17 | 30 | 6 |
| 0 | 0 | 31 | 23 | 20 | 32 |
| 0 | 0 | 24 | 40 | 38 | 10 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,10,33,40,0,0,14,3,15,7,0,0,31,10,11,31,0,0,14,4,27,38],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,11,9,28,28,0,0,32,30,0,13,0,0,0,0,39,37,0,0,0,0,32,2],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,19,4,2,40,0,0,9,4,15,17,0,0,37,0,5,20,0,0,13,9,32,13],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,35,3,31,24,0,0,4,17,23,40,0,0,5,30,20,38,0,0,36,6,32,10] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | 2- (1+4) | Q8.10D10 | D4⋊8D10 |
| kernel | C42.133D10 | C42⋊D5 | C4×D20 | C4.D20 | C42⋊2D5 | C20⋊Q8 | D10.13D4 | C4⋊2D20 | D10⋊Q8 | D10⋊3Q8 | C20.23D4 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{133}D_{10} % in TeX
G:=Group("C4^2.133D10"); // GroupNames label
G:=SmallGroup(320,1254);
// by ID
G=gap.SmallGroup(320,1254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀