metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.92D10, C10.502- (1+4), C20⋊2Q8⋊7C2, (C2×C4).56D20, C4.72(C2×D20), C4⋊C4.269D10, C4.D20⋊4C2, C20.288(C2×D4), (C2×C20).202D4, (C4×C20).8C22, D10⋊2Q8⋊11C2, C42⋊C2⋊10D5, (C2×C10).70C24, C22⋊C4.94D10, C2.16(C22×D20), C10.14(C22×D4), C22.21(C2×D20), (C2×C20).145C23, C22.D20⋊4C2, (C22×C4).191D10, C4⋊Dic5.33C22, C22.99(C23×D5), (C22×Dic10)⋊15C2, (C2×D20).214C22, D10⋊C4.2C22, (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
Subgroups: 974 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8 [×9], C4○D4 [×4], Dic5 [×6], C20 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×10], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×C10, C23.38C23, C4⋊Dic5 [×8], D10⋊C4 [×8], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, C20⋊2Q8 [×2], C4.D20 [×2], C22.D20 [×4], D10⋊2Q8 [×4], C5×C42⋊C2, C22×Dic10, C2×C4○D20, C42.92D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- (1+4) [×2], D20 [×4], C22×D5 [×7], C23.38C23, C2×D20 [×6], C23×D5, C22×D20, D4.10D10 [×2], C42.92D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 138 11 128)(2 139 12 129)(3 140 13 130)(4 121 14 131)(5 122 15 132)(6 123 16 133)(7 124 17 134)(8 125 18 135)(9 126 19 136)(10 127 20 137)(21 50 31 60)(22 51 32 41)(23 52 33 42)(24 53 34 43)(25 54 35 44)(26 55 36 45)(27 56 37 46)(28 57 38 47)(29 58 39 48)(30 59 40 49)(61 99 71 89)(62 100 72 90)(63 81 73 91)(64 82 74 92)(65 83 75 93)(66 84 76 94)(67 85 77 95)(68 86 78 96)(69 87 79 97)(70 88 80 98)(101 160 111 150)(102 141 112 151)(103 142 113 152)(104 143 114 153)(105 144 115 154)(106 145 116 155)(107 146 117 156)(108 147 118 157)(109 148 119 158)(110 149 120 159)
(1 84 155 52)(2 95 156 43)(3 86 157 54)(4 97 158 45)(5 88 159 56)(6 99 160 47)(7 90 141 58)(8 81 142 49)(9 92 143 60)(10 83 144 51)(11 94 145 42)(12 85 146 53)(13 96 147 44)(14 87 148 55)(15 98 149 46)(16 89 150 57)(17 100 151 48)(18 91 152 59)(19 82 153 50)(20 93 154 41)(21 126 64 114)(22 137 65 105)(23 128 66 116)(24 139 67 107)(25 130 68 118)(26 121 69 109)(27 132 70 120)(28 123 71 111)(29 134 72 102)(30 125 73 113)(31 136 74 104)(32 127 75 115)(33 138 76 106)(34 129 77 117)(35 140 78 108)(36 131 79 119)(37 122 80 110)(38 133 61 101)(39 124 62 112)(40 135 63 103)
(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)
(1 137 11 127)(2 126 12 136)(3 135 13 125)(4 124 14 134)(5 133 15 123)(6 122 16 132)(7 131 17 121)(8 140 18 130)(9 129 19 139)(10 138 20 128)(21 85 31 95)(22 94 32 84)(23 83 33 93)(24 92 34 82)(25 81 35 91)(26 90 36 100)(27 99 37 89)(28 88 38 98)(29 97 39 87)(30 86 40 96)(41 66 51 76)(42 75 52 65)(43 64 53 74)(44 73 54 63)(45 62 55 72)(46 71 56 61)(47 80 57 70)(48 69 58 79)(49 78 59 68)(50 67 60 77)(101 149 111 159)(102 158 112 148)(103 147 113 157)(104 156 114 146)(105 145 115 155)(106 154 116 144)(107 143 117 153)(108 152 118 142)(109 141 119 151)(110 150 120 160)
G:=sub<Sym(160)| (1,138,11,128)(2,139,12,129)(3,140,13,130)(4,121,14,131)(5,122,15,132)(6,123,16,133)(7,124,17,134)(8,125,18,135)(9,126,19,136)(10,127,20,137)(21,50,31,60)(22,51,32,41)(23,52,33,42)(24,53,34,43)(25,54,35,44)(26,55,36,45)(27,56,37,46)(28,57,38,47)(29,58,39,48)(30,59,40,49)(61,99,71,89)(62,100,72,90)(63,81,73,91)(64,82,74,92)(65,83,75,93)(66,84,76,94)(67,85,77,95)(68,86,78,96)(69,87,79,97)(70,88,80,98)(101,160,111,150)(102,141,112,151)(103,142,113,152)(104,143,114,153)(105,144,115,154)(106,145,116,155)(107,146,117,156)(108,147,118,157)(109,148,119,158)(110,149,120,159), (1,84,155,52)(2,95,156,43)(3,86,157,54)(4,97,158,45)(5,88,159,56)(6,99,160,47)(7,90,141,58)(8,81,142,49)(9,92,143,60)(10,83,144,51)(11,94,145,42)(12,85,146,53)(13,96,147,44)(14,87,148,55)(15,98,149,46)(16,89,150,57)(17,100,151,48)(18,91,152,59)(19,82,153,50)(20,93,154,41)(21,126,64,114)(22,137,65,105)(23,128,66,116)(24,139,67,107)(25,130,68,118)(26,121,69,109)(27,132,70,120)(28,123,71,111)(29,134,72,102)(30,125,73,113)(31,136,74,104)(32,127,75,115)(33,138,76,106)(34,129,77,117)(35,140,78,108)(36,131,79,119)(37,122,80,110)(38,133,61,101)(39,124,62,112)(40,135,63,103), (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), (1,137,11,127)(2,126,12,136)(3,135,13,125)(4,124,14,134)(5,133,15,123)(6,122,16,132)(7,131,17,121)(8,140,18,130)(9,129,19,139)(10,138,20,128)(21,85,31,95)(22,94,32,84)(23,83,33,93)(24,92,34,82)(25,81,35,91)(26,90,36,100)(27,99,37,89)(28,88,38,98)(29,97,39,87)(30,86,40,96)(41,66,51,76)(42,75,52,65)(43,64,53,74)(44,73,54,63)(45,62,55,72)(46,71,56,61)(47,80,57,70)(48,69,58,79)(49,78,59,68)(50,67,60,77)(101,149,111,159)(102,158,112,148)(103,147,113,157)(104,156,114,146)(105,145,115,155)(106,154,116,144)(107,143,117,153)(108,152,118,142)(109,141,119,151)(110,150,120,160)>;
G:=Group( (1,138,11,128)(2,139,12,129)(3,140,13,130)(4,121,14,131)(5,122,15,132)(6,123,16,133)(7,124,17,134)(8,125,18,135)(9,126,19,136)(10,127,20,137)(21,50,31,60)(22,51,32,41)(23,52,33,42)(24,53,34,43)(25,54,35,44)(26,55,36,45)(27,56,37,46)(28,57,38,47)(29,58,39,48)(30,59,40,49)(61,99,71,89)(62,100,72,90)(63,81,73,91)(64,82,74,92)(65,83,75,93)(66,84,76,94)(67,85,77,95)(68,86,78,96)(69,87,79,97)(70,88,80,98)(101,160,111,150)(102,141,112,151)(103,142,113,152)(104,143,114,153)(105,144,115,154)(106,145,116,155)(107,146,117,156)(108,147,118,157)(109,148,119,158)(110,149,120,159), (1,84,155,52)(2,95,156,43)(3,86,157,54)(4,97,158,45)(5,88,159,56)(6,99,160,47)(7,90,141,58)(8,81,142,49)(9,92,143,60)(10,83,144,51)(11,94,145,42)(12,85,146,53)(13,96,147,44)(14,87,148,55)(15,98,149,46)(16,89,150,57)(17,100,151,48)(18,91,152,59)(19,82,153,50)(20,93,154,41)(21,126,64,114)(22,137,65,105)(23,128,66,116)(24,139,67,107)(25,130,68,118)(26,121,69,109)(27,132,70,120)(28,123,71,111)(29,134,72,102)(30,125,73,113)(31,136,74,104)(32,127,75,115)(33,138,76,106)(34,129,77,117)(35,140,78,108)(36,131,79,119)(37,122,80,110)(38,133,61,101)(39,124,62,112)(40,135,63,103), (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), (1,137,11,127)(2,126,12,136)(3,135,13,125)(4,124,14,134)(5,133,15,123)(6,122,16,132)(7,131,17,121)(8,140,18,130)(9,129,19,139)(10,138,20,128)(21,85,31,95)(22,94,32,84)(23,83,33,93)(24,92,34,82)(25,81,35,91)(26,90,36,100)(27,99,37,89)(28,88,38,98)(29,97,39,87)(30,86,40,96)(41,66,51,76)(42,75,52,65)(43,64,53,74)(44,73,54,63)(45,62,55,72)(46,71,56,61)(47,80,57,70)(48,69,58,79)(49,78,59,68)(50,67,60,77)(101,149,111,159)(102,158,112,148)(103,147,113,157)(104,156,114,146)(105,145,115,155)(106,154,116,144)(107,143,117,153)(108,152,118,142)(109,141,119,151)(110,150,120,160) );
G=PermutationGroup([(1,138,11,128),(2,139,12,129),(3,140,13,130),(4,121,14,131),(5,122,15,132),(6,123,16,133),(7,124,17,134),(8,125,18,135),(9,126,19,136),(10,127,20,137),(21,50,31,60),(22,51,32,41),(23,52,33,42),(24,53,34,43),(25,54,35,44),(26,55,36,45),(27,56,37,46),(28,57,38,47),(29,58,39,48),(30,59,40,49),(61,99,71,89),(62,100,72,90),(63,81,73,91),(64,82,74,92),(65,83,75,93),(66,84,76,94),(67,85,77,95),(68,86,78,96),(69,87,79,97),(70,88,80,98),(101,160,111,150),(102,141,112,151),(103,142,113,152),(104,143,114,153),(105,144,115,154),(106,145,116,155),(107,146,117,156),(108,147,118,157),(109,148,119,158),(110,149,120,159)], [(1,84,155,52),(2,95,156,43),(3,86,157,54),(4,97,158,45),(5,88,159,56),(6,99,160,47),(7,90,141,58),(8,81,142,49),(9,92,143,60),(10,83,144,51),(11,94,145,42),(12,85,146,53),(13,96,147,44),(14,87,148,55),(15,98,149,46),(16,89,150,57),(17,100,151,48),(18,91,152,59),(19,82,153,50),(20,93,154,41),(21,126,64,114),(22,137,65,105),(23,128,66,116),(24,139,67,107),(25,130,68,118),(26,121,69,109),(27,132,70,120),(28,123,71,111),(29,134,72,102),(30,125,73,113),(31,136,74,104),(32,127,75,115),(33,138,76,106),(34,129,77,117),(35,140,78,108),(36,131,79,119),(37,122,80,110),(38,133,61,101),(39,124,62,112),(40,135,63,103)], [(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)], [(1,137,11,127),(2,126,12,136),(3,135,13,125),(4,124,14,134),(5,133,15,123),(6,122,16,132),(7,131,17,121),(8,140,18,130),(9,129,19,139),(10,138,20,128),(21,85,31,95),(22,94,32,84),(23,83,33,93),(24,92,34,82),(25,81,35,91),(26,90,36,100),(27,99,37,89),(28,88,38,98),(29,97,39,87),(30,86,40,96),(41,66,51,76),(42,75,52,65),(43,64,53,74),(44,73,54,63),(45,62,55,72),(46,71,56,61),(47,80,57,70),(48,69,58,79),(49,78,59,68),(50,67,60,77),(101,149,111,159),(102,158,112,148),(103,147,113,157),(104,156,114,146),(105,145,115,155),(106,154,116,144),(107,143,117,153),(108,152,118,142),(109,141,119,151),(110,150,120,160)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀