metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.233D10, (C4×D5)⋊6D4, C4.30(D4×D5), C20.59(C2×D4), (D5×C42)⋊8C2, C20⋊D4⋊23C2, D10.18(C2×D4), C4.4D4⋊18D5, D10⋊D4⋊37C2, Dic5⋊3(C4○D4), (C2×D4).168D10, C4.D20⋊22C2, (C2×C20).76C23, (C2×Q8).134D10, C22⋊C4.70D10, C10.86(C22×D4), Dic5⋊Q8⋊19C2, Dic5⋊4D4⋊27C2, (C4×C20).182C22, (C2×C10).212C24, Dic5.120(C2×D4), C23.34(C22×D5), (D4×C10).150C22, (C2×D20).166C22, (C22×C10).42C23, C5⋊4(C22.26C24), (Q8×C10).121C22, C22.233(C23×D5), D10⋊C4.58C22, (C2×Dic5).259C23, (C4×Dic5).338C22, C10.D4.47C22, (C22×D5).222C23, (C2×Dic10).179C22, (C22×Dic5).137C22, C2.59(C2×D4×D5), C2.71(D5×C4○D4), (C2×Q8⋊2D5)⋊9C2, (C5×C4.4D4)⋊6C2, (C2×D4⋊2D5)⋊18C2, C10.183(C2×C4○D4), (C2×C4×D5).127C22, (C2×C4).298(C22×D5), (C2×C5⋊D4).55C22, (C5×C22⋊C4).59C22, SmallGroup(320,1340)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1166 in 310 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×12], C22, C22 [×16], C5, C2×C4, C2×C4 [×4], C2×C4 [×21], D4 [×20], Q8 [×4], C23 [×2], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×7], C2×D4, C2×D4 [×9], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×8], C2×C10, C2×C10 [×6], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×2], C4×D5 [×4], C4×D5 [×8], D20 [×6], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10 [×2], C22.26C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×4], D10⋊C4 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], D4⋊2D5 [×4], Q8⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×6], D4×C10, Q8×C10, D5×C42, C4.D20, Dic5⋊4D4 [×4], D10⋊D4 [×4], C20⋊D4, Dic5⋊Q8, C5×C4.4D4, C2×D4⋊2D5, C2×Q8⋊2D5, C42.233D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4 [×2], C42.233D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c-1 >
►On 160 points
Generators in S160
(1 135 33 108)(2 114 34 84)(3 137 35 110)(4 116 36 86)(5 139 37 102)(6 118 38 88)(7 131 39 104)(8 120 40 90)(9 133 31 106)(10 112 32 82)(11 119 41 89)(12 132 42 105)(13 111 43 81)(14 134 44 107)(15 113 45 83)(16 136 46 109)(17 115 47 85)(18 138 48 101)(19 117 49 87)(20 140 50 103)(21 95 143 55)(22 79 144 64)(23 97 145 57)(24 71 146 66)(25 99 147 59)(26 73 148 68)(27 91 149 51)(28 75 150 70)(29 93 141 53)(30 77 142 62)(52 155 92 122)(54 157 94 124)(56 159 96 126)(58 151 98 128)(60 153 100 130)(61 156 76 123)(63 158 78 125)(65 160 80 127)(67 152 72 129)(69 154 74 121)
(1 100 15 73)(2 51 16 69)(3 92 17 75)(4 53 18 61)(5 94 19 77)(6 55 20 63)(7 96 11 79)(8 57 12 65)(9 98 13 71)(10 59 14 67)(21 140 158 118)(22 104 159 89)(23 132 160 120)(24 106 151 81)(25 134 152 112)(26 108 153 83)(27 136 154 114)(28 110 155 85)(29 138 156 116)(30 102 157 87)(31 58 43 66)(32 99 44 72)(33 60 45 68)(34 91 46 74)(35 52 47 70)(36 93 48 76)(37 54 49 62)(38 95 50 78)(39 56 41 64)(40 97 42 80)(82 147 107 129)(84 149 109 121)(86 141 101 123)(88 143 103 125)(90 145 105 127)(111 146 133 128)(113 148 135 130)(115 150 137 122)(117 142 139 124)(119 144 131 126)
(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)
(1 124)(2 156)(3 122)(4 154)(5 130)(6 152)(7 128)(8 160)(9 126)(10 158)(11 146)(12 23)(13 144)(14 21)(15 142)(16 29)(17 150)(18 27)(19 148)(20 25)(22 43)(24 41)(26 49)(28 47)(30 45)(31 159)(32 125)(33 157)(34 123)(35 155)(36 121)(37 153)(38 129)(39 151)(40 127)(42 145)(44 143)(46 141)(48 149)(50 147)(51 86)(52 115)(53 84)(54 113)(55 82)(56 111)(57 90)(58 119)(59 88)(60 117)(61 109)(62 135)(63 107)(64 133)(65 105)(66 131)(67 103)(68 139)(69 101)(70 137)(71 104)(72 140)(73 102)(74 138)(75 110)(76 136)(77 108)(78 134)(79 106)(80 132)(81 96)(83 94)(85 92)(87 100)(89 98)(91 116)(93 114)(95 112)(97 120)(99 118)
G:=sub<Sym(160)| (1,135,33,108)(2,114,34,84)(3,137,35,110)(4,116,36,86)(5,139,37,102)(6,118,38,88)(7,131,39,104)(8,120,40,90)(9,133,31,106)(10,112,32,82)(11,119,41,89)(12,132,42,105)(13,111,43,81)(14,134,44,107)(15,113,45,83)(16,136,46,109)(17,115,47,85)(18,138,48,101)(19,117,49,87)(20,140,50,103)(21,95,143,55)(22,79,144,64)(23,97,145,57)(24,71,146,66)(25,99,147,59)(26,73,148,68)(27,91,149,51)(28,75,150,70)(29,93,141,53)(30,77,142,62)(52,155,92,122)(54,157,94,124)(56,159,96,126)(58,151,98,128)(60,153,100,130)(61,156,76,123)(63,158,78,125)(65,160,80,127)(67,152,72,129)(69,154,74,121), (1,100,15,73)(2,51,16,69)(3,92,17,75)(4,53,18,61)(5,94,19,77)(6,55,20,63)(7,96,11,79)(8,57,12,65)(9,98,13,71)(10,59,14,67)(21,140,158,118)(22,104,159,89)(23,132,160,120)(24,106,151,81)(25,134,152,112)(26,108,153,83)(27,136,154,114)(28,110,155,85)(29,138,156,116)(30,102,157,87)(31,58,43,66)(32,99,44,72)(33,60,45,68)(34,91,46,74)(35,52,47,70)(36,93,48,76)(37,54,49,62)(38,95,50,78)(39,56,41,64)(40,97,42,80)(82,147,107,129)(84,149,109,121)(86,141,101,123)(88,143,103,125)(90,145,105,127)(111,146,133,128)(113,148,135,130)(115,150,137,122)(117,142,139,124)(119,144,131,126), (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), (1,124)(2,156)(3,122)(4,154)(5,130)(6,152)(7,128)(8,160)(9,126)(10,158)(11,146)(12,23)(13,144)(14,21)(15,142)(16,29)(17,150)(18,27)(19,148)(20,25)(22,43)(24,41)(26,49)(28,47)(30,45)(31,159)(32,125)(33,157)(34,123)(35,155)(36,121)(37,153)(38,129)(39,151)(40,127)(42,145)(44,143)(46,141)(48,149)(50,147)(51,86)(52,115)(53,84)(54,113)(55,82)(56,111)(57,90)(58,119)(59,88)(60,117)(61,109)(62,135)(63,107)(64,133)(65,105)(66,131)(67,103)(68,139)(69,101)(70,137)(71,104)(72,140)(73,102)(74,138)(75,110)(76,136)(77,108)(78,134)(79,106)(80,132)(81,96)(83,94)(85,92)(87,100)(89,98)(91,116)(93,114)(95,112)(97,120)(99,118)>;
G:=Group( (1,135,33,108)(2,114,34,84)(3,137,35,110)(4,116,36,86)(5,139,37,102)(6,118,38,88)(7,131,39,104)(8,120,40,90)(9,133,31,106)(10,112,32,82)(11,119,41,89)(12,132,42,105)(13,111,43,81)(14,134,44,107)(15,113,45,83)(16,136,46,109)(17,115,47,85)(18,138,48,101)(19,117,49,87)(20,140,50,103)(21,95,143,55)(22,79,144,64)(23,97,145,57)(24,71,146,66)(25,99,147,59)(26,73,148,68)(27,91,149,51)(28,75,150,70)(29,93,141,53)(30,77,142,62)(52,155,92,122)(54,157,94,124)(56,159,96,126)(58,151,98,128)(60,153,100,130)(61,156,76,123)(63,158,78,125)(65,160,80,127)(67,152,72,129)(69,154,74,121), (1,100,15,73)(2,51,16,69)(3,92,17,75)(4,53,18,61)(5,94,19,77)(6,55,20,63)(7,96,11,79)(8,57,12,65)(9,98,13,71)(10,59,14,67)(21,140,158,118)(22,104,159,89)(23,132,160,120)(24,106,151,81)(25,134,152,112)(26,108,153,83)(27,136,154,114)(28,110,155,85)(29,138,156,116)(30,102,157,87)(31,58,43,66)(32,99,44,72)(33,60,45,68)(34,91,46,74)(35,52,47,70)(36,93,48,76)(37,54,49,62)(38,95,50,78)(39,56,41,64)(40,97,42,80)(82,147,107,129)(84,149,109,121)(86,141,101,123)(88,143,103,125)(90,145,105,127)(111,146,133,128)(113,148,135,130)(115,150,137,122)(117,142,139,124)(119,144,131,126), (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), (1,124)(2,156)(3,122)(4,154)(5,130)(6,152)(7,128)(8,160)(9,126)(10,158)(11,146)(12,23)(13,144)(14,21)(15,142)(16,29)(17,150)(18,27)(19,148)(20,25)(22,43)(24,41)(26,49)(28,47)(30,45)(31,159)(32,125)(33,157)(34,123)(35,155)(36,121)(37,153)(38,129)(39,151)(40,127)(42,145)(44,143)(46,141)(48,149)(50,147)(51,86)(52,115)(53,84)(54,113)(55,82)(56,111)(57,90)(58,119)(59,88)(60,117)(61,109)(62,135)(63,107)(64,133)(65,105)(66,131)(67,103)(68,139)(69,101)(70,137)(71,104)(72,140)(73,102)(74,138)(75,110)(76,136)(77,108)(78,134)(79,106)(80,132)(81,96)(83,94)(85,92)(87,100)(89,98)(91,116)(93,114)(95,112)(97,120)(99,118) );
G=PermutationGroup([(1,135,33,108),(2,114,34,84),(3,137,35,110),(4,116,36,86),(5,139,37,102),(6,118,38,88),(7,131,39,104),(8,120,40,90),(9,133,31,106),(10,112,32,82),(11,119,41,89),(12,132,42,105),(13,111,43,81),(14,134,44,107),(15,113,45,83),(16,136,46,109),(17,115,47,85),(18,138,48,101),(19,117,49,87),(20,140,50,103),(21,95,143,55),(22,79,144,64),(23,97,145,57),(24,71,146,66),(25,99,147,59),(26,73,148,68),(27,91,149,51),(28,75,150,70),(29,93,141,53),(30,77,142,62),(52,155,92,122),(54,157,94,124),(56,159,96,126),(58,151,98,128),(60,153,100,130),(61,156,76,123),(63,158,78,125),(65,160,80,127),(67,152,72,129),(69,154,74,121)], [(1,100,15,73),(2,51,16,69),(3,92,17,75),(4,53,18,61),(5,94,19,77),(6,55,20,63),(7,96,11,79),(8,57,12,65),(9,98,13,71),(10,59,14,67),(21,140,158,118),(22,104,159,89),(23,132,160,120),(24,106,151,81),(25,134,152,112),(26,108,153,83),(27,136,154,114),(28,110,155,85),(29,138,156,116),(30,102,157,87),(31,58,43,66),(32,99,44,72),(33,60,45,68),(34,91,46,74),(35,52,47,70),(36,93,48,76),(37,54,49,62),(38,95,50,78),(39,56,41,64),(40,97,42,80),(82,147,107,129),(84,149,109,121),(86,141,101,123),(88,143,103,125),(90,145,105,127),(111,146,133,128),(113,148,135,130),(115,150,137,122),(117,142,139,124),(119,144,131,126)], [(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)], [(1,124),(2,156),(3,122),(4,154),(5,130),(6,152),(7,128),(8,160),(9,126),(10,158),(11,146),(12,23),(13,144),(14,21),(15,142),(16,29),(17,150),(18,27),(19,148),(20,25),(22,43),(24,41),(26,49),(28,47),(30,45),(31,159),(32,125),(33,157),(34,123),(35,155),(36,121),(37,153),(38,129),(39,151),(40,127),(42,145),(44,143),(46,141),(48,149),(50,147),(51,86),(52,115),(53,84),(54,113),(55,82),(56,111),(57,90),(58,119),(59,88),(60,117),(61,109),(62,135),(63,107),(64,133),(65,105),(66,131),(67,103),(68,139),(69,101),(70,137),(71,104),(72,140),(73,102),(74,138),(75,110),(76,136),(77,108),(78,134),(79,106),(80,132),(81,96),(83,94),(85,92),(87,100),(89,98),(91,116),(93,114),(95,112),(97,120),(99,118)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 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 | 1 | 4 |
| 0 | 0 | 0 | 0 | 20 | 40 |
| 9 | 18 | 0 | 0 | 0 | 0 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 37 |
| 0 | 0 | 0 | 0 | 21 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 37 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 20 | 40 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,20,0,0,0,0,4,40],[9,32,0,0,0,0,18,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,21,0,0,0,0,37,1],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,37,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,20,0,0,0,0,0,40] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D4×D5 | D5×C4○D4 |
| kernel | C42.233D10 | D5×C42 | C4.D20 | Dic5⋊4D4 | D10⋊D4 | C20⋊D4 | Dic5⋊Q8 | C5×C4.4D4 | C2×D4⋊2D5 | C2×Q8⋊2D5 | C4×D5 | C4.4D4 | Dic5 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 8 | 2 | 8 | 2 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{233}D_{10} % in TeX
G:=Group("C4^2.233D10"); // GroupNames label
G:=SmallGroup(320,1340);
// by ID
G=gap.SmallGroup(320,1340);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,232,100,1123,346,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀