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⋊D20.12C2, Dic5⋊3Q8⋊35C2, D10.16(C4○D4), C20.128(C4○D4), (C2×C10).235C24, (C2×C20).506C23, (C4×C20).195C22, 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
Generators and relations for C42.237D10
G = < a,b,c,d | a4=b4=d2=1, c10=a2, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c9 >
Subgroups: 830 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C23.36C23, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, D5×C42, C4.D20, Dic5⋊3Q8, C4⋊C4⋊7D5, D20⋊8C4, D20⋊8C4, D10.13D4, C4⋊D20, D10⋊2Q8, C4⋊C4⋊D5, C5×C42.C2, C42.237D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, Q8⋊2D5, C23×D5, C2×Q8⋊2D5, D5×C4○D4, C42.237D10
►On 160 points
Generators in S160
(1 108 11 118)(2 72 12 62)(3 110 13 120)(4 74 14 64)(5 112 15 102)(6 76 16 66)(7 114 17 104)(8 78 18 68)(9 116 19 106)(10 80 20 70)(21 140 31 130)(22 158 32 148)(23 122 33 132)(24 160 34 150)(25 124 35 134)(26 142 36 152)(27 126 37 136)(28 144 38 154)(29 128 39 138)(30 146 40 156)(41 157 51 147)(42 121 52 131)(43 159 53 149)(44 123 54 133)(45 141 55 151)(46 125 56 135)(47 143 57 153)(48 127 58 137)(49 145 59 155)(50 129 60 139)(61 89 71 99)(63 91 73 81)(65 93 75 83)(67 95 77 85)(69 97 79 87)(82 101 92 111)(84 103 94 113)(86 105 96 115)(88 107 98 117)(90 109 100 119)
(1 126 89 143)(2 137 90 154)(3 128 91 145)(4 139 92 156)(5 130 93 147)(6 121 94 158)(7 132 95 149)(8 123 96 160)(9 134 97 151)(10 125 98 142)(11 136 99 153)(12 127 100 144)(13 138 81 155)(14 129 82 146)(15 140 83 157)(16 131 84 148)(17 122 85 159)(18 133 86 150)(19 124 87 141)(20 135 88 152)(21 75 41 112)(22 66 42 103)(23 77 43 114)(24 68 44 105)(25 79 45 116)(26 70 46 107)(27 61 47 118)(28 72 48 109)(29 63 49 120)(30 74 50 111)(31 65 51 102)(32 76 52 113)(33 67 53 104)(34 78 54 115)(35 69 55 106)(36 80 56 117)(37 71 57 108)(38 62 58 119)(39 73 59 110)(40 64 60 101)
(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 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(61 102)(62 101)(63 120)(64 119)(65 118)(66 117)(67 116)(68 115)(69 114)(70 113)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
G:=sub<Sym(160)| (1,108,11,118)(2,72,12,62)(3,110,13,120)(4,74,14,64)(5,112,15,102)(6,76,16,66)(7,114,17,104)(8,78,18,68)(9,116,19,106)(10,80,20,70)(21,140,31,130)(22,158,32,148)(23,122,33,132)(24,160,34,150)(25,124,35,134)(26,142,36,152)(27,126,37,136)(28,144,38,154)(29,128,39,138)(30,146,40,156)(41,157,51,147)(42,121,52,131)(43,159,53,149)(44,123,54,133)(45,141,55,151)(46,125,56,135)(47,143,57,153)(48,127,58,137)(49,145,59,155)(50,129,60,139)(61,89,71,99)(63,91,73,81)(65,93,75,83)(67,95,77,85)(69,97,79,87)(82,101,92,111)(84,103,94,113)(86,105,96,115)(88,107,98,117)(90,109,100,119), (1,126,89,143)(2,137,90,154)(3,128,91,145)(4,139,92,156)(5,130,93,147)(6,121,94,158)(7,132,95,149)(8,123,96,160)(9,134,97,151)(10,125,98,142)(11,136,99,153)(12,127,100,144)(13,138,81,155)(14,129,82,146)(15,140,83,157)(16,131,84,148)(17,122,85,159)(18,133,86,150)(19,124,87,141)(20,135,88,152)(21,75,41,112)(22,66,42,103)(23,77,43,114)(24,68,44,105)(25,79,45,116)(26,70,46,107)(27,61,47,118)(28,72,48,109)(29,63,49,120)(30,74,50,111)(31,65,51,102)(32,76,52,113)(33,67,53,104)(34,78,54,115)(35,69,55,106)(36,80,56,117)(37,71,57,108)(38,62,58,119)(39,73,59,110)(40,64,60,101), (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)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(61,102)(62,101)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)>;
G:=Group( (1,108,11,118)(2,72,12,62)(3,110,13,120)(4,74,14,64)(5,112,15,102)(6,76,16,66)(7,114,17,104)(8,78,18,68)(9,116,19,106)(10,80,20,70)(21,140,31,130)(22,158,32,148)(23,122,33,132)(24,160,34,150)(25,124,35,134)(26,142,36,152)(27,126,37,136)(28,144,38,154)(29,128,39,138)(30,146,40,156)(41,157,51,147)(42,121,52,131)(43,159,53,149)(44,123,54,133)(45,141,55,151)(46,125,56,135)(47,143,57,153)(48,127,58,137)(49,145,59,155)(50,129,60,139)(61,89,71,99)(63,91,73,81)(65,93,75,83)(67,95,77,85)(69,97,79,87)(82,101,92,111)(84,103,94,113)(86,105,96,115)(88,107,98,117)(90,109,100,119), (1,126,89,143)(2,137,90,154)(3,128,91,145)(4,139,92,156)(5,130,93,147)(6,121,94,158)(7,132,95,149)(8,123,96,160)(9,134,97,151)(10,125,98,142)(11,136,99,153)(12,127,100,144)(13,138,81,155)(14,129,82,146)(15,140,83,157)(16,131,84,148)(17,122,85,159)(18,133,86,150)(19,124,87,141)(20,135,88,152)(21,75,41,112)(22,66,42,103)(23,77,43,114)(24,68,44,105)(25,79,45,116)(26,70,46,107)(27,61,47,118)(28,72,48,109)(29,63,49,120)(30,74,50,111)(31,65,51,102)(32,76,52,113)(33,67,53,104)(34,78,54,115)(35,69,55,106)(36,80,56,117)(37,71,57,108)(38,62,58,119)(39,73,59,110)(40,64,60,101), (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)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(61,102)(62,101)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(121,125)(122,124)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151) );
G=PermutationGroup([[(1,108,11,118),(2,72,12,62),(3,110,13,120),(4,74,14,64),(5,112,15,102),(6,76,16,66),(7,114,17,104),(8,78,18,68),(9,116,19,106),(10,80,20,70),(21,140,31,130),(22,158,32,148),(23,122,33,132),(24,160,34,150),(25,124,35,134),(26,142,36,152),(27,126,37,136),(28,144,38,154),(29,128,39,138),(30,146,40,156),(41,157,51,147),(42,121,52,131),(43,159,53,149),(44,123,54,133),(45,141,55,151),(46,125,56,135),(47,143,57,153),(48,127,58,137),(49,145,59,155),(50,129,60,139),(61,89,71,99),(63,91,73,81),(65,93,75,83),(67,95,77,85),(69,97,79,87),(82,101,92,111),(84,103,94,113),(86,105,96,115),(88,107,98,117),(90,109,100,119)], [(1,126,89,143),(2,137,90,154),(3,128,91,145),(4,139,92,156),(5,130,93,147),(6,121,94,158),(7,132,95,149),(8,123,96,160),(9,134,97,151),(10,125,98,142),(11,136,99,153),(12,127,100,144),(13,138,81,155),(14,129,82,146),(15,140,83,157),(16,131,84,148),(17,122,85,159),(18,133,86,150),(19,124,87,141),(20,135,88,152),(21,75,41,112),(22,66,42,103),(23,77,43,114),(24,68,44,105),(25,79,45,116),(26,70,46,107),(27,61,47,118),(28,72,48,109),(29,63,49,120),(30,74,50,111),(31,65,51,102),(32,76,52,113),(33,67,53,104),(34,78,54,115),(35,69,55,106),(36,80,56,117),(37,71,57,108),(38,62,58,119),(39,73,59,110),(40,64,60,101)], [(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),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(61,102),(62,101),(63,120),(64,119),(65,118),(66,117),(67,116),(68,115),(69,114),(70,113),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(121,125),(122,124),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151)]])
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⋊D20 | 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 |
Matrix representation of C42.237D10 ►in 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] >;
C42.237D10 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