metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.99D10, C10.542- (1+4), C10.992+ (1+4), (C4×D20)⋊11C2, C20⋊2Q8⋊8C2, C4⋊C4.274D10, D10⋊Q8⋊6C2, C4.D20⋊5C2, (C4×Dic10)⋊12C2, C20⋊7D4.18C2, C42⋊C2⋊18D5, (C2×C10).78C24, (C4×C20).29C22, D10.12D4⋊5C2, C20.236(C4○D4), C4.120(C4○D20), C20.48D4⋊42C2, (C2×C20).151C23, C22⋊C4.102D10, Dic5.5D4⋊5C2, (C22×C4).199D10, C4⋊Dic5.36C22, C2.11(D4⋊8D10), C23.89(C22×D5), C23.D5.6C22, (C2×D20).216C22, D10⋊C4.4C22, (C2×Dic5).31C23, (C22×D5).26C23, C22.107(C23×D5), (C22×C10).148C23, (C22×C20).308C22, C5⋊1(C22.36C24), (C4×Dic5).220C22, C10.D4.75C22, C2.12(D4.10D10), (C2×Dic10).241C22, C4⋊C4⋊D5⋊6C2, C2.37(C2×C4○D20), C10.34(C2×C4○D4), (C2×C4×D5).246C22, (C5×C42⋊C2)⋊20C2, (C5×C4⋊C4).314C22, (C2×C4).151(C22×D5), (C2×C5⋊D4).11C22, (C5×C22⋊C4).117C22, SmallGroup(320,1206)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 782 in 216 conjugacy classes, 95 normal (51 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D5 [×2], C10 [×3], C10, C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8 [×3], Dic5 [×6], C20 [×2], C20 [×5], D10 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×4], C4×D5 [×2], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×2], C2×C20 [×6], C2×C20 [×2], C22×D5 [×2], C22×C10, C22.36C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C4×Dic10, C20⋊2Q8, C4×D20, C4.D20, D10.12D4 [×2], Dic5.5D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], C20.48D4, C20⋊7D4, C5×C42⋊C2, C42.99D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.36C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊8D10, D4.10D10, C42.99D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >
►On 160 points
Generators in S160
(1 113 33 98)(2 114 34 99)(3 115 35 100)(4 116 36 91)(5 117 37 92)(6 118 38 93)(7 119 39 94)(8 120 40 95)(9 111 31 96)(10 112 32 97)(11 128 48 109)(12 129 49 110)(13 130 50 101)(14 121 41 102)(15 122 42 103)(16 123 43 104)(17 124 44 105)(18 125 45 106)(19 126 46 107)(20 127 47 108)(21 85 135 55)(22 86 136 56)(23 87 137 57)(24 88 138 58)(25 89 139 59)(26 90 140 60)(27 81 131 51)(28 82 132 52)(29 83 133 53)(30 84 134 54)(61 156 76 141)(62 157 77 142)(63 158 78 143)(64 159 79 144)(65 160 80 145)(66 151 71 146)(67 152 72 147)(68 153 73 148)(69 154 74 149)(70 155 75 150)
(1 90 18 73)(2 51 19 69)(3 82 20 75)(4 53 11 61)(5 84 12 77)(6 55 13 63)(7 86 14 79)(8 57 15 65)(9 88 16 71)(10 59 17 67)(21 130 158 118)(22 102 159 94)(23 122 160 120)(24 104 151 96)(25 124 152 112)(26 106 153 98)(27 126 154 114)(28 108 155 100)(29 128 156 116)(30 110 157 92)(31 58 43 66)(32 89 44 72)(33 60 45 68)(34 81 46 74)(35 52 47 70)(36 83 48 76)(37 54 49 62)(38 85 50 78)(39 56 41 64)(40 87 42 80)(91 133 109 141)(93 135 101 143)(95 137 103 145)(97 139 105 147)(99 131 107 149)(111 138 123 146)(113 140 125 148)(115 132 127 150)(117 134 129 142)(119 136 121 144)
(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 10 18 17)(2 16 19 9)(3 8 20 15)(4 14 11 7)(5 6 12 13)(21 142 158 134)(22 133 159 141)(23 150 160 132)(24 131 151 149)(25 148 152 140)(26 139 153 147)(27 146 154 138)(28 137 155 145)(29 144 156 136)(30 135 157 143)(31 34 43 46)(32 45 44 33)(35 40 47 42)(36 41 48 39)(37 38 49 50)(51 66 69 58)(52 57 70 65)(53 64 61 56)(54 55 62 63)(59 68 67 60)(71 74 88 81)(72 90 89 73)(75 80 82 87)(76 86 83 79)(77 78 84 85)(91 121 109 119)(92 118 110 130)(93 129 101 117)(94 116 102 128)(95 127 103 115)(96 114 104 126)(97 125 105 113)(98 112 106 124)(99 123 107 111)(100 120 108 122)
G:=sub<Sym(160)| (1,113,33,98)(2,114,34,99)(3,115,35,100)(4,116,36,91)(5,117,37,92)(6,118,38,93)(7,119,39,94)(8,120,40,95)(9,111,31,96)(10,112,32,97)(11,128,48,109)(12,129,49,110)(13,130,50,101)(14,121,41,102)(15,122,42,103)(16,123,43,104)(17,124,44,105)(18,125,45,106)(19,126,46,107)(20,127,47,108)(21,85,135,55)(22,86,136,56)(23,87,137,57)(24,88,138,58)(25,89,139,59)(26,90,140,60)(27,81,131,51)(28,82,132,52)(29,83,133,53)(30,84,134,54)(61,156,76,141)(62,157,77,142)(63,158,78,143)(64,159,79,144)(65,160,80,145)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,90,18,73)(2,51,19,69)(3,82,20,75)(4,53,11,61)(5,84,12,77)(6,55,13,63)(7,86,14,79)(8,57,15,65)(9,88,16,71)(10,59,17,67)(21,130,158,118)(22,102,159,94)(23,122,160,120)(24,104,151,96)(25,124,152,112)(26,106,153,98)(27,126,154,114)(28,108,155,100)(29,128,156,116)(30,110,157,92)(31,58,43,66)(32,89,44,72)(33,60,45,68)(34,81,46,74)(35,52,47,70)(36,83,48,76)(37,54,49,62)(38,85,50,78)(39,56,41,64)(40,87,42,80)(91,133,109,141)(93,135,101,143)(95,137,103,145)(97,139,105,147)(99,131,107,149)(111,138,123,146)(113,140,125,148)(115,132,127,150)(117,134,129,142)(119,136,121,144), (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), (1,10,18,17)(2,16,19,9)(3,8,20,15)(4,14,11,7)(5,6,12,13)(21,142,158,134)(22,133,159,141)(23,150,160,132)(24,131,151,149)(25,148,152,140)(26,139,153,147)(27,146,154,138)(28,137,155,145)(29,144,156,136)(30,135,157,143)(31,34,43,46)(32,45,44,33)(35,40,47,42)(36,41,48,39)(37,38,49,50)(51,66,69,58)(52,57,70,65)(53,64,61,56)(54,55,62,63)(59,68,67,60)(71,74,88,81)(72,90,89,73)(75,80,82,87)(76,86,83,79)(77,78,84,85)(91,121,109,119)(92,118,110,130)(93,129,101,117)(94,116,102,128)(95,127,103,115)(96,114,104,126)(97,125,105,113)(98,112,106,124)(99,123,107,111)(100,120,108,122)>;
G:=Group( (1,113,33,98)(2,114,34,99)(3,115,35,100)(4,116,36,91)(5,117,37,92)(6,118,38,93)(7,119,39,94)(8,120,40,95)(9,111,31,96)(10,112,32,97)(11,128,48,109)(12,129,49,110)(13,130,50,101)(14,121,41,102)(15,122,42,103)(16,123,43,104)(17,124,44,105)(18,125,45,106)(19,126,46,107)(20,127,47,108)(21,85,135,55)(22,86,136,56)(23,87,137,57)(24,88,138,58)(25,89,139,59)(26,90,140,60)(27,81,131,51)(28,82,132,52)(29,83,133,53)(30,84,134,54)(61,156,76,141)(62,157,77,142)(63,158,78,143)(64,159,79,144)(65,160,80,145)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,90,18,73)(2,51,19,69)(3,82,20,75)(4,53,11,61)(5,84,12,77)(6,55,13,63)(7,86,14,79)(8,57,15,65)(9,88,16,71)(10,59,17,67)(21,130,158,118)(22,102,159,94)(23,122,160,120)(24,104,151,96)(25,124,152,112)(26,106,153,98)(27,126,154,114)(28,108,155,100)(29,128,156,116)(30,110,157,92)(31,58,43,66)(32,89,44,72)(33,60,45,68)(34,81,46,74)(35,52,47,70)(36,83,48,76)(37,54,49,62)(38,85,50,78)(39,56,41,64)(40,87,42,80)(91,133,109,141)(93,135,101,143)(95,137,103,145)(97,139,105,147)(99,131,107,149)(111,138,123,146)(113,140,125,148)(115,132,127,150)(117,134,129,142)(119,136,121,144), (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), (1,10,18,17)(2,16,19,9)(3,8,20,15)(4,14,11,7)(5,6,12,13)(21,142,158,134)(22,133,159,141)(23,150,160,132)(24,131,151,149)(25,148,152,140)(26,139,153,147)(27,146,154,138)(28,137,155,145)(29,144,156,136)(30,135,157,143)(31,34,43,46)(32,45,44,33)(35,40,47,42)(36,41,48,39)(37,38,49,50)(51,66,69,58)(52,57,70,65)(53,64,61,56)(54,55,62,63)(59,68,67,60)(71,74,88,81)(72,90,89,73)(75,80,82,87)(76,86,83,79)(77,78,84,85)(91,121,109,119)(92,118,110,130)(93,129,101,117)(94,116,102,128)(95,127,103,115)(96,114,104,126)(97,125,105,113)(98,112,106,124)(99,123,107,111)(100,120,108,122) );
G=PermutationGroup([(1,113,33,98),(2,114,34,99),(3,115,35,100),(4,116,36,91),(5,117,37,92),(6,118,38,93),(7,119,39,94),(8,120,40,95),(9,111,31,96),(10,112,32,97),(11,128,48,109),(12,129,49,110),(13,130,50,101),(14,121,41,102),(15,122,42,103),(16,123,43,104),(17,124,44,105),(18,125,45,106),(19,126,46,107),(20,127,47,108),(21,85,135,55),(22,86,136,56),(23,87,137,57),(24,88,138,58),(25,89,139,59),(26,90,140,60),(27,81,131,51),(28,82,132,52),(29,83,133,53),(30,84,134,54),(61,156,76,141),(62,157,77,142),(63,158,78,143),(64,159,79,144),(65,160,80,145),(66,151,71,146),(67,152,72,147),(68,153,73,148),(69,154,74,149),(70,155,75,150)], [(1,90,18,73),(2,51,19,69),(3,82,20,75),(4,53,11,61),(5,84,12,77),(6,55,13,63),(7,86,14,79),(8,57,15,65),(9,88,16,71),(10,59,17,67),(21,130,158,118),(22,102,159,94),(23,122,160,120),(24,104,151,96),(25,124,152,112),(26,106,153,98),(27,126,154,114),(28,108,155,100),(29,128,156,116),(30,110,157,92),(31,58,43,66),(32,89,44,72),(33,60,45,68),(34,81,46,74),(35,52,47,70),(36,83,48,76),(37,54,49,62),(38,85,50,78),(39,56,41,64),(40,87,42,80),(91,133,109,141),(93,135,101,143),(95,137,103,145),(97,139,105,147),(99,131,107,149),(111,138,123,146),(113,140,125,148),(115,132,127,150),(117,134,129,142),(119,136,121,144)], [(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)], [(1,10,18,17),(2,16,19,9),(3,8,20,15),(4,14,11,7),(5,6,12,13),(21,142,158,134),(22,133,159,141),(23,150,160,132),(24,131,151,149),(25,148,152,140),(26,139,153,147),(27,146,154,138),(28,137,155,145),(29,144,156,136),(30,135,157,143),(31,34,43,46),(32,45,44,33),(35,40,47,42),(36,41,48,39),(37,38,49,50),(51,66,69,58),(52,57,70,65),(53,64,61,56),(54,55,62,63),(59,68,67,60),(71,74,88,81),(72,90,89,73),(75,80,82,87),(76,86,83,79),(77,78,84,85),(91,121,109,119),(92,118,110,130),(93,129,101,117),(94,116,102,128),(95,127,103,115),(96,114,104,126),(97,125,105,113),(98,112,106,124),(99,123,107,111),(100,120,108,122)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 5 | 22 |
| 0 | 0 | 0 | 14 | 19 | 36 |
| 0 | 0 | 36 | 19 | 27 | 0 |
| 0 | 0 | 22 | 5 | 0 | 27 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,14,0,36,22,0,0,0,14,19,5,0,0,5,19,27,0,0,0,22,36,0,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,40,0,0,40,34,0,0,0,0,0,1,0,0] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | ··· | 2 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | 2- (1+4) | D4⋊8D10 | D4.10D10 |
| kernel | C42.99D10 | C4×Dic10 | C20⋊2Q8 | C4×D20 | C4.D20 | D10.12D4 | Dic5.5D4 | D10⋊Q8 | C4⋊C4⋊D5 | C20.48D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{99}D_{10} % in TeX
G:=Group("C4^2.99D10"); // GroupNames label
G:=SmallGroup(320,1206);
// by ID
G=gap.SmallGroup(320,1206);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀