metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.92D10, C10.502- 1+4, C20⋊2Q8⋊7C2, C4.72(C2×D20), (C2×C4).56D20, C4⋊C4.269D10, (C2×C20).202D4, C4.D20⋊4C2, C20.288(C2×D4), (C4×C20).8C22, D10⋊2Q8⋊11C2, C42⋊C2⋊10D5, (C2×C10).70C24, C22⋊C4.94D10, C22.21(C2×D20), C10.14(C22×D4), C2.16(C22×D20), (C2×C20).145C23, C22.D20⋊4C2, (C22×C4).191D10, C4⋊Dic5.33C22, C22.99(C23×D5), (C22×Dic10)⋊15C2, D10⋊C4.2C22, (C2×D20).214C22, (C2×Dic5).24C23, (C22×D5).20C23, C23.158(C22×D5), C2.8(D4.10D10), (C22×C10).140C23, (C22×C20).230C22, C5⋊1(C23.38C23), (C2×Dic10).293C22, (C22×Dic5).87C22, (C2×C10).51(C2×D4), (C2×C4×D5).68C22, (C2×C4○D20).19C2, (C5×C42⋊C2)⋊12C2, (C5×C4⋊C4).307C22, (C2×C4).576(C22×D5), (C2×C5⋊D4).109C22, (C5×C22⋊C4).102C22, SmallGroup(320,1198)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.92D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, dbd-1=b-1, dcd-1=c9 >
Subgroups: 974 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.38C23, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C20⋊2Q8, C4.D20, C22.D20, D10⋊2Q8, C5×C42⋊C2, C22×Dic10, C2×C4○D20, C42.92D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2- 1+4, D20, C22×D5, C23.38C23, C2×D20, C23×D5, C22×D20, D4.10D10, C42.92D10
►On 160 points
Generators in S160
(1 157 11 147)(2 158 12 148)(3 159 13 149)(4 160 14 150)(5 141 15 151)(6 142 16 152)(7 143 17 153)(8 144 18 154)(9 145 19 155)(10 146 20 156)(21 121 31 131)(22 122 32 132)(23 123 33 133)(24 124 34 134)(25 125 35 135)(26 126 36 136)(27 127 37 137)(28 128 38 138)(29 129 39 139)(30 130 40 140)(41 93 51 83)(42 94 52 84)(43 95 53 85)(44 96 54 86)(45 97 55 87)(46 98 56 88)(47 99 57 89)(48 100 58 90)(49 81 59 91)(50 82 60 92)(61 103 71 113)(62 104 72 114)(63 105 73 115)(64 106 74 116)(65 107 75 117)(66 108 76 118)(67 109 77 119)(68 110 78 120)(69 111 79 101)(70 112 80 102)
(1 119 130 45)(2 110 131 56)(3 101 132 47)(4 112 133 58)(5 103 134 49)(6 114 135 60)(7 105 136 51)(8 116 137 42)(9 107 138 53)(10 118 139 44)(11 109 140 55)(12 120 121 46)(13 111 122 57)(14 102 123 48)(15 113 124 59)(16 104 125 50)(17 115 126 41)(18 106 127 52)(19 117 128 43)(20 108 129 54)(21 88 158 78)(22 99 159 69)(23 90 160 80)(24 81 141 71)(25 92 142 62)(26 83 143 73)(27 94 144 64)(28 85 145 75)(29 96 146 66)(30 87 147 77)(31 98 148 68)(32 89 149 79)(33 100 150 70)(34 91 151 61)(35 82 152 72)(36 93 153 63)(37 84 154 74)(38 95 155 65)(39 86 156 76)(40 97 157 67)
(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)
(1 156 11 146)(2 145 12 155)(3 154 13 144)(4 143 14 153)(5 152 15 142)(6 141 16 151)(7 150 17 160)(8 159 18 149)(9 148 19 158)(10 157 20 147)(21 138 31 128)(22 127 32 137)(23 136 33 126)(24 125 34 135)(25 134 35 124)(26 123 36 133)(27 132 37 122)(28 121 38 131)(29 130 39 140)(30 139 40 129)(41 80 51 70)(42 69 52 79)(43 78 53 68)(44 67 54 77)(45 76 55 66)(46 65 56 75)(47 74 57 64)(48 63 58 73)(49 72 59 62)(50 61 60 71)(81 104 91 114)(82 113 92 103)(83 102 93 112)(84 111 94 101)(85 120 95 110)(86 109 96 119)(87 118 97 108)(88 107 98 117)(89 116 99 106)(90 105 100 115)
G:=sub<Sym(160)| (1,157,11,147)(2,158,12,148)(3,159,13,149)(4,160,14,150)(5,141,15,151)(6,142,16,152)(7,143,17,153)(8,144,18,154)(9,145,19,155)(10,146,20,156)(21,121,31,131)(22,122,32,132)(23,123,33,133)(24,124,34,134)(25,125,35,135)(26,126,36,136)(27,127,37,137)(28,128,38,138)(29,129,39,139)(30,130,40,140)(41,93,51,83)(42,94,52,84)(43,95,53,85)(44,96,54,86)(45,97,55,87)(46,98,56,88)(47,99,57,89)(48,100,58,90)(49,81,59,91)(50,82,60,92)(61,103,71,113)(62,104,72,114)(63,105,73,115)(64,106,74,116)(65,107,75,117)(66,108,76,118)(67,109,77,119)(68,110,78,120)(69,111,79,101)(70,112,80,102), (1,119,130,45)(2,110,131,56)(3,101,132,47)(4,112,133,58)(5,103,134,49)(6,114,135,60)(7,105,136,51)(8,116,137,42)(9,107,138,53)(10,118,139,44)(11,109,140,55)(12,120,121,46)(13,111,122,57)(14,102,123,48)(15,113,124,59)(16,104,125,50)(17,115,126,41)(18,106,127,52)(19,117,128,43)(20,108,129,54)(21,88,158,78)(22,99,159,69)(23,90,160,80)(24,81,141,71)(25,92,142,62)(26,83,143,73)(27,94,144,64)(28,85,145,75)(29,96,146,66)(30,87,147,77)(31,98,148,68)(32,89,149,79)(33,100,150,70)(34,91,151,61)(35,82,152,72)(36,93,153,63)(37,84,154,74)(38,95,155,65)(39,86,156,76)(40,97,157,67), (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), (1,156,11,146)(2,145,12,155)(3,154,13,144)(4,143,14,153)(5,152,15,142)(6,141,16,151)(7,150,17,160)(8,159,18,149)(9,148,19,158)(10,157,20,147)(21,138,31,128)(22,127,32,137)(23,136,33,126)(24,125,34,135)(25,134,35,124)(26,123,36,133)(27,132,37,122)(28,121,38,131)(29,130,39,140)(30,139,40,129)(41,80,51,70)(42,69,52,79)(43,78,53,68)(44,67,54,77)(45,76,55,66)(46,65,56,75)(47,74,57,64)(48,63,58,73)(49,72,59,62)(50,61,60,71)(81,104,91,114)(82,113,92,103)(83,102,93,112)(84,111,94,101)(85,120,95,110)(86,109,96,119)(87,118,97,108)(88,107,98,117)(89,116,99,106)(90,105,100,115)>;
G:=Group( (1,157,11,147)(2,158,12,148)(3,159,13,149)(4,160,14,150)(5,141,15,151)(6,142,16,152)(7,143,17,153)(8,144,18,154)(9,145,19,155)(10,146,20,156)(21,121,31,131)(22,122,32,132)(23,123,33,133)(24,124,34,134)(25,125,35,135)(26,126,36,136)(27,127,37,137)(28,128,38,138)(29,129,39,139)(30,130,40,140)(41,93,51,83)(42,94,52,84)(43,95,53,85)(44,96,54,86)(45,97,55,87)(46,98,56,88)(47,99,57,89)(48,100,58,90)(49,81,59,91)(50,82,60,92)(61,103,71,113)(62,104,72,114)(63,105,73,115)(64,106,74,116)(65,107,75,117)(66,108,76,118)(67,109,77,119)(68,110,78,120)(69,111,79,101)(70,112,80,102), (1,119,130,45)(2,110,131,56)(3,101,132,47)(4,112,133,58)(5,103,134,49)(6,114,135,60)(7,105,136,51)(8,116,137,42)(9,107,138,53)(10,118,139,44)(11,109,140,55)(12,120,121,46)(13,111,122,57)(14,102,123,48)(15,113,124,59)(16,104,125,50)(17,115,126,41)(18,106,127,52)(19,117,128,43)(20,108,129,54)(21,88,158,78)(22,99,159,69)(23,90,160,80)(24,81,141,71)(25,92,142,62)(26,83,143,73)(27,94,144,64)(28,85,145,75)(29,96,146,66)(30,87,147,77)(31,98,148,68)(32,89,149,79)(33,100,150,70)(34,91,151,61)(35,82,152,72)(36,93,153,63)(37,84,154,74)(38,95,155,65)(39,86,156,76)(40,97,157,67), (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), (1,156,11,146)(2,145,12,155)(3,154,13,144)(4,143,14,153)(5,152,15,142)(6,141,16,151)(7,150,17,160)(8,159,18,149)(9,148,19,158)(10,157,20,147)(21,138,31,128)(22,127,32,137)(23,136,33,126)(24,125,34,135)(25,134,35,124)(26,123,36,133)(27,132,37,122)(28,121,38,131)(29,130,39,140)(30,139,40,129)(41,80,51,70)(42,69,52,79)(43,78,53,68)(44,67,54,77)(45,76,55,66)(46,65,56,75)(47,74,57,64)(48,63,58,73)(49,72,59,62)(50,61,60,71)(81,104,91,114)(82,113,92,103)(83,102,93,112)(84,111,94,101)(85,120,95,110)(86,109,96,119)(87,118,97,108)(88,107,98,117)(89,116,99,106)(90,105,100,115) );
G=PermutationGroup([[(1,157,11,147),(2,158,12,148),(3,159,13,149),(4,160,14,150),(5,141,15,151),(6,142,16,152),(7,143,17,153),(8,144,18,154),(9,145,19,155),(10,146,20,156),(21,121,31,131),(22,122,32,132),(23,123,33,133),(24,124,34,134),(25,125,35,135),(26,126,36,136),(27,127,37,137),(28,128,38,138),(29,129,39,139),(30,130,40,140),(41,93,51,83),(42,94,52,84),(43,95,53,85),(44,96,54,86),(45,97,55,87),(46,98,56,88),(47,99,57,89),(48,100,58,90),(49,81,59,91),(50,82,60,92),(61,103,71,113),(62,104,72,114),(63,105,73,115),(64,106,74,116),(65,107,75,117),(66,108,76,118),(67,109,77,119),(68,110,78,120),(69,111,79,101),(70,112,80,102)], [(1,119,130,45),(2,110,131,56),(3,101,132,47),(4,112,133,58),(5,103,134,49),(6,114,135,60),(7,105,136,51),(8,116,137,42),(9,107,138,53),(10,118,139,44),(11,109,140,55),(12,120,121,46),(13,111,122,57),(14,102,123,48),(15,113,124,59),(16,104,125,50),(17,115,126,41),(18,106,127,52),(19,117,128,43),(20,108,129,54),(21,88,158,78),(22,99,159,69),(23,90,160,80),(24,81,141,71),(25,92,142,62),(26,83,143,73),(27,94,144,64),(28,85,145,75),(29,96,146,66),(30,87,147,77),(31,98,148,68),(32,89,149,79),(33,100,150,70),(34,91,151,61),(35,82,152,72),(36,93,153,63),(37,84,154,74),(38,95,155,65),(39,86,156,76),(40,97,157,67)], [(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)], [(1,156,11,146),(2,145,12,155),(3,154,13,144),(4,143,14,153),(5,152,15,142),(6,141,16,151),(7,150,17,160),(8,159,18,149),(9,148,19,158),(10,157,20,147),(21,138,31,128),(22,127,32,137),(23,136,33,126),(24,125,34,135),(25,134,35,124),(26,123,36,133),(27,132,37,122),(28,121,38,131),(29,130,39,140),(30,139,40,129),(41,80,51,70),(42,69,52,79),(43,78,53,68),(44,67,54,77),(45,76,55,66),(46,65,56,75),(47,74,57,64),(48,63,58,73),(49,72,59,62),(50,61,60,71),(81,104,91,114),(82,113,92,103),(83,102,93,112),(84,111,94,101),(85,120,95,110),(86,109,96,119),(87,118,97,108),(88,107,98,117),(89,116,99,106),(90,105,100,115)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 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 | D4 | D5 | D10 | D10 | D10 | D10 | D20 | 2- 1+4 | D4.10D10 |
| kernel | C42.92D10 | C20⋊2Q8 | C4.D20 | C22.D20 | D10⋊2Q8 | C5×C42⋊C2 | C22×Dic10 | C2×C4○D20 | C2×C20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
Matrix representation of C42.92D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 9 | 0 | 0 |
| 0 | 0 | 32 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 9 |
| 0 | 0 | 0 | 0 | 32 | 11 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 9 | 39 | 0 |
| 0 | 0 | 32 | 11 | 0 | 39 |
| 0 | 0 | 0 | 0 | 11 | 32 |
| 0 | 0 | 0 | 0 | 9 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 25 | 35 | 1 |
| 0 | 0 | 16 | 23 | 40 | 1 |
| 0 | 0 | 12 | 14 | 29 | 16 |
| 0 | 0 | 27 | 28 | 25 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 16 | 25 | 28 |
| 0 | 0 | 13 | 18 | 39 | 16 |
| 0 | 0 | 0 | 0 | 12 | 25 |
| 0 | 0 | 0 | 0 | 27 | 29 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,39,0,11,9,0,0,0,39,32,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,16,12,27,0,0,25,23,14,28,0,0,35,40,29,25,0,0,1,1,16,18],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,23,13,0,0,0,0,16,18,0,0,0,0,25,39,12,27,0,0,28,16,25,29] >;
C42.92D10 in GAP, Magma, Sage, TeX
C_4^2._{92}D_{10} % in TeX
G:=Group("C4^2.92D10"); // GroupNames label
G:=SmallGroup(320,1198);
// by ID
G=gap.SmallGroup(320,1198);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations