metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.237D10, C4⋊C4.206D10, (D5×C42)⋊11C2, C42.C2⋊16D5, D20⋊8C4⋊35C2, D10⋊2Q8⋊36C2, C4.D20⋊24C2, C4⋊2D20.12C2, Dic5⋊3Q8⋊35C2, D10.16(C4○D4), C20.128(C4○D4), (C4×C20).195C22, (C2×C10).235C24, (C2×C20).506C23, C4.19(Q8⋊2D5), D10.13D4⋊33C2, Dic5.75(C4○D4), (C2×D20).169C22, C4⋊Dic5.241C22, C22.256(C23×D5), D10⋊C4.60C22, (C2×Dic5).379C23, (C4×Dic5).150C22, C10.D4.51C22, (C22×D5).101C23, C5⋊10(C23.36C23), (C2×Dic10).185C22, C2.86(D5×C4○D4), C4⋊C4⋊7D5⋊35C2, C4⋊C4⋊D5⋊33C2, (C5×C42.C2)⋊8C2, C10.197(C2×C4○D4), C2.22(C2×Q8⋊2D5), (C2×C4×D5).134C22, (C2×C4).79(C22×D5), (C5×C4⋊C4).190C22, SmallGroup(320,1363)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 830 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10 [×3], C42, C42 [×5], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×8], C2×C10, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C22×D5, C22×D5 [×2], C23.36C23, C4×Dic5 [×3], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×3], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], D5×C42, C4.D20, Dic5⋊3Q8, C4⋊C4⋊7D5 [×2], D20⋊8C4, D20⋊8C4 [×2], D10.13D4 [×2], C4⋊2D20, D10⋊2Q8, C4⋊C4⋊D5 [×2], C5×C42.C2, C42.237D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, Q8⋊2D5 [×2], C23×D5, C2×Q8⋊2D5, D5×C4○D4 [×2], C42.237D10
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c10=a2, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c9 >
►On 160 points
Generators in S160
(1 60 11 50)(2 126 12 136)(3 42 13 52)(4 128 14 138)(5 44 15 54)(6 130 16 140)(7 46 17 56)(8 132 18 122)(9 48 19 58)(10 134 20 124)(21 139 31 129)(22 55 32 45)(23 121 33 131)(24 57 34 47)(25 123 35 133)(26 59 36 49)(27 125 37 135)(28 41 38 51)(29 127 39 137)(30 43 40 53)(61 153 71 143)(62 84 72 94)(63 155 73 145)(64 86 74 96)(65 157 75 147)(66 88 76 98)(67 159 77 149)(68 90 78 100)(69 141 79 151)(70 92 80 82)(81 108 91 118)(83 110 93 120)(85 112 95 102)(87 114 97 104)(89 116 99 106)(101 154 111 144)(103 156 113 146)(105 158 115 148)(107 160 117 150)(109 142 119 152)
(1 67 27 106)(2 78 28 117)(3 69 29 108)(4 80 30 119)(5 71 31 110)(6 62 32 101)(7 73 33 112)(8 64 34 103)(9 75 35 114)(10 66 36 105)(11 77 37 116)(12 68 38 107)(13 79 39 118)(14 70 40 109)(15 61 21 120)(16 72 22 111)(17 63 23 102)(18 74 24 113)(19 65 25 104)(20 76 26 115)(41 150 126 100)(42 141 127 91)(43 152 128 82)(44 143 129 93)(45 154 130 84)(46 145 131 95)(47 156 132 86)(48 147 133 97)(49 158 134 88)(50 149 135 99)(51 160 136 90)(52 151 137 81)(53 142 138 92)(54 153 139 83)(55 144 140 94)(56 155 121 85)(57 146 122 96)(58 157 123 87)(59 148 124 98)(60 159 125 89)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 128)(42 127)(43 126)(44 125)(45 124)(46 123)(47 122)(48 121)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 141)(82 160)(83 159)(84 158)(85 157)(86 156)(87 155)(88 154)(89 153)(90 152)(91 151)(92 150)(93 149)(94 148)(95 147)(96 146)(97 145)(98 144)(99 143)(100 142)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
G:=sub<Sym(160)| (1,60,11,50)(2,126,12,136)(3,42,13,52)(4,128,14,138)(5,44,15,54)(6,130,16,140)(7,46,17,56)(8,132,18,122)(9,48,19,58)(10,134,20,124)(21,139,31,129)(22,55,32,45)(23,121,33,131)(24,57,34,47)(25,123,35,133)(26,59,36,49)(27,125,37,135)(28,41,38,51)(29,127,39,137)(30,43,40,53)(61,153,71,143)(62,84,72,94)(63,155,73,145)(64,86,74,96)(65,157,75,147)(66,88,76,98)(67,159,77,149)(68,90,78,100)(69,141,79,151)(70,92,80,82)(81,108,91,118)(83,110,93,120)(85,112,95,102)(87,114,97,104)(89,116,99,106)(101,154,111,144)(103,156,113,146)(105,158,115,148)(107,160,117,150)(109,142,119,152), (1,67,27,106)(2,78,28,117)(3,69,29,108)(4,80,30,119)(5,71,31,110)(6,62,32,101)(7,73,33,112)(8,64,34,103)(9,75,35,114)(10,66,36,105)(11,77,37,116)(12,68,38,107)(13,79,39,118)(14,70,40,109)(15,61,21,120)(16,72,22,111)(17,63,23,102)(18,74,24,113)(19,65,25,104)(20,76,26,115)(41,150,126,100)(42,141,127,91)(43,152,128,82)(44,143,129,93)(45,154,130,84)(46,145,131,95)(47,156,132,86)(48,147,133,97)(49,158,134,88)(50,149,135,99)(51,160,136,90)(52,151,137,81)(53,142,138,92)(54,153,139,83)(55,144,140,94)(56,155,121,85)(57,146,122,96)(58,157,123,87)(59,148,124,98)(60,159,125,89), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,141)(82,160)(83,159)(84,158)(85,157)(86,156)(87,155)(88,154)(89,153)(90,152)(91,151)(92,150)(93,149)(94,148)(95,147)(96,146)(97,145)(98,144)(99,143)(100,142)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)>;
G:=Group( (1,60,11,50)(2,126,12,136)(3,42,13,52)(4,128,14,138)(5,44,15,54)(6,130,16,140)(7,46,17,56)(8,132,18,122)(9,48,19,58)(10,134,20,124)(21,139,31,129)(22,55,32,45)(23,121,33,131)(24,57,34,47)(25,123,35,133)(26,59,36,49)(27,125,37,135)(28,41,38,51)(29,127,39,137)(30,43,40,53)(61,153,71,143)(62,84,72,94)(63,155,73,145)(64,86,74,96)(65,157,75,147)(66,88,76,98)(67,159,77,149)(68,90,78,100)(69,141,79,151)(70,92,80,82)(81,108,91,118)(83,110,93,120)(85,112,95,102)(87,114,97,104)(89,116,99,106)(101,154,111,144)(103,156,113,146)(105,158,115,148)(107,160,117,150)(109,142,119,152), (1,67,27,106)(2,78,28,117)(3,69,29,108)(4,80,30,119)(5,71,31,110)(6,62,32,101)(7,73,33,112)(8,64,34,103)(9,75,35,114)(10,66,36,105)(11,77,37,116)(12,68,38,107)(13,79,39,118)(14,70,40,109)(15,61,21,120)(16,72,22,111)(17,63,23,102)(18,74,24,113)(19,65,25,104)(20,76,26,115)(41,150,126,100)(42,141,127,91)(43,152,128,82)(44,143,129,93)(45,154,130,84)(46,145,131,95)(47,156,132,86)(48,147,133,97)(49,158,134,88)(50,149,135,99)(51,160,136,90)(52,151,137,81)(53,142,138,92)(54,153,139,83)(55,144,140,94)(56,155,121,85)(57,146,122,96)(58,157,123,87)(59,148,124,98)(60,159,125,89), (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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(81,141)(82,160)(83,159)(84,158)(85,157)(86,156)(87,155)(88,154)(89,153)(90,152)(91,151)(92,150)(93,149)(94,148)(95,147)(96,146)(97,145)(98,144)(99,143)(100,142)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114) );
G=PermutationGroup([(1,60,11,50),(2,126,12,136),(3,42,13,52),(4,128,14,138),(5,44,15,54),(6,130,16,140),(7,46,17,56),(8,132,18,122),(9,48,19,58),(10,134,20,124),(21,139,31,129),(22,55,32,45),(23,121,33,131),(24,57,34,47),(25,123,35,133),(26,59,36,49),(27,125,37,135),(28,41,38,51),(29,127,39,137),(30,43,40,53),(61,153,71,143),(62,84,72,94),(63,155,73,145),(64,86,74,96),(65,157,75,147),(66,88,76,98),(67,159,77,149),(68,90,78,100),(69,141,79,151),(70,92,80,82),(81,108,91,118),(83,110,93,120),(85,112,95,102),(87,114,97,104),(89,116,99,106),(101,154,111,144),(103,156,113,146),(105,158,115,148),(107,160,117,150),(109,142,119,152)], [(1,67,27,106),(2,78,28,117),(3,69,29,108),(4,80,30,119),(5,71,31,110),(6,62,32,101),(7,73,33,112),(8,64,34,103),(9,75,35,114),(10,66,36,105),(11,77,37,116),(12,68,38,107),(13,79,39,118),(14,70,40,109),(15,61,21,120),(16,72,22,111),(17,63,23,102),(18,74,24,113),(19,65,25,104),(20,76,26,115),(41,150,126,100),(42,141,127,91),(43,152,128,82),(44,143,129,93),(45,154,130,84),(46,145,131,95),(47,156,132,86),(48,147,133,97),(49,158,134,88),(50,149,135,99),(51,160,136,90),(52,151,137,81),(53,142,138,92),(54,153,139,83),(55,144,140,94),(56,155,121,85),(57,146,122,96),(58,157,123,87),(59,148,124,98),(60,159,125,89)], [(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40),(41,128),(42,127),(43,126),(44,125),(45,124),(46,123),(47,122),(48,121),(49,140),(50,139),(51,138),(52,137),(53,136),(54,135),(55,134),(56,133),(57,132),(58,131),(59,130),(60,129),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(81,141),(82,160),(83,159),(84,158),(85,157),(86,156),(87,155),(88,154),(89,153),(90,152),(91,151),(92,150),(93,149),(94,148),(95,147),(96,146),(97,145),(98,144),(99,143),(100,142),(101,105),(102,104),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 33 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,33,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,1,10,0,0,0,0,0,40,0,0,0,0,0,0,1,35,0,0,0,0,6,6],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,10,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,0,1] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | C4○D4 | D10 | D10 | Q8⋊2D5 | D5×C4○D4 |
| kernel | C42.237D10 | D5×C42 | C4.D20 | Dic5⋊3Q8 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | Dic5 | C20 | D10 | C42 | C4⋊C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 12 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{237}D_{10} % in TeX
G:=Group("C4^2.237D10"); // GroupNames label
G:=SmallGroup(320,1363);
// by ID
G=gap.SmallGroup(320,1363);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,346,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=a^2,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^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀