metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.91D10, C10.942+ (1+4), C10.492- (1+4), (C4×D20)⋊6C2, C4○D20⋊17C4, D20⋊37(C2×C4), C4⋊C4.310D10, (C4×Dic10)⋊8C2, C42⋊C2⋊7D5, Dic10⋊34(C2×C4), C10.40(C23×C4), (C4×C20).23C22, (C2×C10).67C24, Dic5⋊4D4⋊42C2, C2.2(D4⋊8D10), C20.149(C22×C4), (C2×C20).489C23, C22⋊C4.127D10, D10.15(C22×C4), (C22×C4).189D10, C22.29(C23×D5), (C2×D20).294C22, C4⋊Dic5.397C22, Dic5.16(C22×C4), C23.155(C22×D5), C2.2(D4.10D10), (C22×C10).137C23, (C22×C20).227C22, C5⋊3(C23.33C23), (C4×Dic5).215C22, (C2×Dic5).206C23, (C22×D5).174C23, D10⋊C4.119C22, (C2×Dic10).323C22, C10.D4.132C22, (C22×Dic5).86C22, (C2×C4)⋊7(C4×D5), C4.94(C2×C4×D5), (D5×C4⋊C4)⋊11C2, (C4×D5)⋊2(C2×C4), (C2×C20)⋊26(C2×C4), C5⋊D4⋊11(C2×C4), C22.7(C2×C4×D5), C4⋊C4⋊7D5⋊11C2, C2.21(D5×C22×C4), (C2×C4⋊Dic5)⋊39C2, (C2×C4×D5).67C22, (C5×C42⋊C2)⋊9C2, (C2×C4○D20).18C2, (C5×C4⋊C4).306C22, (C2×C4).273(C22×D5), (C2×C10).124(C22×C4), (C2×C5⋊D4).106C22, (C5×C22⋊C4).137C22, SmallGroup(320,1195)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 926 in 294 conjugacy classes, 151 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10 [×3], C10 [×2], C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], C4×D5 [×4], D20 [×4], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×C10, C23.33C23, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×4], D10⋊C4 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, C4×Dic10 [×2], C4×D20 [×2], Dic5⋊4D4 [×4], D5×C4⋊C4 [×2], C4⋊C4⋊7D5 [×2], C2×C4⋊Dic5, C5×C42⋊C2, C2×C4○D20, C42.91D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ (1+4), 2- (1+4), C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D4⋊8D10, D4.10D10, C42.91D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=c9 >
►On 160 points
Generators in S160
(1 152 64 38)(2 143 65 29)(3 154 66 40)(4 145 67 31)(5 156 68 22)(6 147 69 33)(7 158 70 24)(8 149 71 35)(9 160 72 26)(10 151 73 37)(11 142 74 28)(12 153 75 39)(13 144 76 30)(14 155 77 21)(15 146 78 32)(16 157 79 23)(17 148 80 34)(18 159 61 25)(19 150 62 36)(20 141 63 27)(41 101 83 128)(42 112 84 139)(43 103 85 130)(44 114 86 121)(45 105 87 132)(46 116 88 123)(47 107 89 134)(48 118 90 125)(49 109 91 136)(50 120 92 127)(51 111 93 138)(52 102 94 129)(53 113 95 140)(54 104 96 131)(55 115 97 122)(56 106 98 133)(57 117 99 124)(58 108 100 135)(59 119 81 126)(60 110 82 137)
(1 51 74 83)(2 42 75 94)(3 53 76 85)(4 44 77 96)(5 55 78 87)(6 46 79 98)(7 57 80 89)(8 48 61 100)(9 59 62 91)(10 50 63 82)(11 41 64 93)(12 52 65 84)(13 43 66 95)(14 54 67 86)(15 45 68 97)(16 56 69 88)(17 47 70 99)(18 58 71 90)(19 49 72 81)(20 60 73 92)(21 131 145 114)(22 122 146 105)(23 133 147 116)(24 124 148 107)(25 135 149 118)(26 126 150 109)(27 137 151 120)(28 128 152 111)(29 139 153 102)(30 130 154 113)(31 121 155 104)(32 132 156 115)(33 123 157 106)(34 134 158 117)(35 125 159 108)(36 136 160 119)(37 127 141 110)(38 138 142 101)(39 129 143 112)(40 140 144 103)
(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)
(1 120 11 110)(2 109 12 119)(3 118 13 108)(4 107 14 117)(5 116 15 106)(6 105 16 115)(7 114 17 104)(8 103 18 113)(9 112 19 102)(10 101 20 111)(21 47 31 57)(22 56 32 46)(23 45 33 55)(24 54 34 44)(25 43 35 53)(26 52 36 42)(27 41 37 51)(28 50 38 60)(29 59 39 49)(30 48 40 58)(61 140 71 130)(62 129 72 139)(63 138 73 128)(64 127 74 137)(65 136 75 126)(66 125 76 135)(67 134 77 124)(68 123 78 133)(69 132 79 122)(70 121 80 131)(81 153 91 143)(82 142 92 152)(83 151 93 141)(84 160 94 150)(85 149 95 159)(86 158 96 148)(87 147 97 157)(88 156 98 146)(89 145 99 155)(90 154 100 144)
G:=sub<Sym(160)| (1,152,64,38)(2,143,65,29)(3,154,66,40)(4,145,67,31)(5,156,68,22)(6,147,69,33)(7,158,70,24)(8,149,71,35)(9,160,72,26)(10,151,73,37)(11,142,74,28)(12,153,75,39)(13,144,76,30)(14,155,77,21)(15,146,78,32)(16,157,79,23)(17,148,80,34)(18,159,61,25)(19,150,62,36)(20,141,63,27)(41,101,83,128)(42,112,84,139)(43,103,85,130)(44,114,86,121)(45,105,87,132)(46,116,88,123)(47,107,89,134)(48,118,90,125)(49,109,91,136)(50,120,92,127)(51,111,93,138)(52,102,94,129)(53,113,95,140)(54,104,96,131)(55,115,97,122)(56,106,98,133)(57,117,99,124)(58,108,100,135)(59,119,81,126)(60,110,82,137), (1,51,74,83)(2,42,75,94)(3,53,76,85)(4,44,77,96)(5,55,78,87)(6,46,79,98)(7,57,80,89)(8,48,61,100)(9,59,62,91)(10,50,63,82)(11,41,64,93)(12,52,65,84)(13,43,66,95)(14,54,67,86)(15,45,68,97)(16,56,69,88)(17,47,70,99)(18,58,71,90)(19,49,72,81)(20,60,73,92)(21,131,145,114)(22,122,146,105)(23,133,147,116)(24,124,148,107)(25,135,149,118)(26,126,150,109)(27,137,151,120)(28,128,152,111)(29,139,153,102)(30,130,154,113)(31,121,155,104)(32,132,156,115)(33,123,157,106)(34,134,158,117)(35,125,159,108)(36,136,160,119)(37,127,141,110)(38,138,142,101)(39,129,143,112)(40,140,144,103), (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), (1,120,11,110)(2,109,12,119)(3,118,13,108)(4,107,14,117)(5,116,15,106)(6,105,16,115)(7,114,17,104)(8,103,18,113)(9,112,19,102)(10,101,20,111)(21,47,31,57)(22,56,32,46)(23,45,33,55)(24,54,34,44)(25,43,35,53)(26,52,36,42)(27,41,37,51)(28,50,38,60)(29,59,39,49)(30,48,40,58)(61,140,71,130)(62,129,72,139)(63,138,73,128)(64,127,74,137)(65,136,75,126)(66,125,76,135)(67,134,77,124)(68,123,78,133)(69,132,79,122)(70,121,80,131)(81,153,91,143)(82,142,92,152)(83,151,93,141)(84,160,94,150)(85,149,95,159)(86,158,96,148)(87,147,97,157)(88,156,98,146)(89,145,99,155)(90,154,100,144)>;
G:=Group( (1,152,64,38)(2,143,65,29)(3,154,66,40)(4,145,67,31)(5,156,68,22)(6,147,69,33)(7,158,70,24)(8,149,71,35)(9,160,72,26)(10,151,73,37)(11,142,74,28)(12,153,75,39)(13,144,76,30)(14,155,77,21)(15,146,78,32)(16,157,79,23)(17,148,80,34)(18,159,61,25)(19,150,62,36)(20,141,63,27)(41,101,83,128)(42,112,84,139)(43,103,85,130)(44,114,86,121)(45,105,87,132)(46,116,88,123)(47,107,89,134)(48,118,90,125)(49,109,91,136)(50,120,92,127)(51,111,93,138)(52,102,94,129)(53,113,95,140)(54,104,96,131)(55,115,97,122)(56,106,98,133)(57,117,99,124)(58,108,100,135)(59,119,81,126)(60,110,82,137), (1,51,74,83)(2,42,75,94)(3,53,76,85)(4,44,77,96)(5,55,78,87)(6,46,79,98)(7,57,80,89)(8,48,61,100)(9,59,62,91)(10,50,63,82)(11,41,64,93)(12,52,65,84)(13,43,66,95)(14,54,67,86)(15,45,68,97)(16,56,69,88)(17,47,70,99)(18,58,71,90)(19,49,72,81)(20,60,73,92)(21,131,145,114)(22,122,146,105)(23,133,147,116)(24,124,148,107)(25,135,149,118)(26,126,150,109)(27,137,151,120)(28,128,152,111)(29,139,153,102)(30,130,154,113)(31,121,155,104)(32,132,156,115)(33,123,157,106)(34,134,158,117)(35,125,159,108)(36,136,160,119)(37,127,141,110)(38,138,142,101)(39,129,143,112)(40,140,144,103), (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), (1,120,11,110)(2,109,12,119)(3,118,13,108)(4,107,14,117)(5,116,15,106)(6,105,16,115)(7,114,17,104)(8,103,18,113)(9,112,19,102)(10,101,20,111)(21,47,31,57)(22,56,32,46)(23,45,33,55)(24,54,34,44)(25,43,35,53)(26,52,36,42)(27,41,37,51)(28,50,38,60)(29,59,39,49)(30,48,40,58)(61,140,71,130)(62,129,72,139)(63,138,73,128)(64,127,74,137)(65,136,75,126)(66,125,76,135)(67,134,77,124)(68,123,78,133)(69,132,79,122)(70,121,80,131)(81,153,91,143)(82,142,92,152)(83,151,93,141)(84,160,94,150)(85,149,95,159)(86,158,96,148)(87,147,97,157)(88,156,98,146)(89,145,99,155)(90,154,100,144) );
G=PermutationGroup([(1,152,64,38),(2,143,65,29),(3,154,66,40),(4,145,67,31),(5,156,68,22),(6,147,69,33),(7,158,70,24),(8,149,71,35),(9,160,72,26),(10,151,73,37),(11,142,74,28),(12,153,75,39),(13,144,76,30),(14,155,77,21),(15,146,78,32),(16,157,79,23),(17,148,80,34),(18,159,61,25),(19,150,62,36),(20,141,63,27),(41,101,83,128),(42,112,84,139),(43,103,85,130),(44,114,86,121),(45,105,87,132),(46,116,88,123),(47,107,89,134),(48,118,90,125),(49,109,91,136),(50,120,92,127),(51,111,93,138),(52,102,94,129),(53,113,95,140),(54,104,96,131),(55,115,97,122),(56,106,98,133),(57,117,99,124),(58,108,100,135),(59,119,81,126),(60,110,82,137)], [(1,51,74,83),(2,42,75,94),(3,53,76,85),(4,44,77,96),(5,55,78,87),(6,46,79,98),(7,57,80,89),(8,48,61,100),(9,59,62,91),(10,50,63,82),(11,41,64,93),(12,52,65,84),(13,43,66,95),(14,54,67,86),(15,45,68,97),(16,56,69,88),(17,47,70,99),(18,58,71,90),(19,49,72,81),(20,60,73,92),(21,131,145,114),(22,122,146,105),(23,133,147,116),(24,124,148,107),(25,135,149,118),(26,126,150,109),(27,137,151,120),(28,128,152,111),(29,139,153,102),(30,130,154,113),(31,121,155,104),(32,132,156,115),(33,123,157,106),(34,134,158,117),(35,125,159,108),(36,136,160,119),(37,127,141,110),(38,138,142,101),(39,129,143,112),(40,140,144,103)], [(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)], [(1,120,11,110),(2,109,12,119),(3,118,13,108),(4,107,14,117),(5,116,15,106),(6,105,16,115),(7,114,17,104),(8,103,18,113),(9,112,19,102),(10,101,20,111),(21,47,31,57),(22,56,32,46),(23,45,33,55),(24,54,34,44),(25,43,35,53),(26,52,36,42),(27,41,37,51),(28,50,38,60),(29,59,39,49),(30,48,40,58),(61,140,71,130),(62,129,72,139),(63,138,73,128),(64,127,74,137),(65,136,75,126),(66,125,76,135),(67,134,77,124),(68,123,78,133),(69,132,79,122),(70,121,80,131),(81,153,91,143),(82,142,92,152),(83,151,93,141),(84,160,94,150),(85,149,95,159),(86,158,96,148),(87,147,97,157),(88,156,98,146),(89,145,99,155),(90,154,100,144)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 34 | 34 |
| 0 | 0 | 0 | 4 | 14 | 1 |
| 0 | 0 | 30 | 5 | 37 | 0 |
| 0 | 0 | 31 | 36 | 0 | 37 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 30 | 18 | 23 |
| 0 | 0 | 6 | 3 | 0 | 21 |
| 0 | 0 | 26 | 34 | 33 | 11 |
| 0 | 0 | 0 | 34 | 30 | 8 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 3 | 1 |
| 0 | 0 | 35 | 0 | 37 | 37 |
| 0 | 0 | 17 | 21 | 5 | 30 |
| 0 | 0 | 13 | 34 | 11 | 30 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 10 | 22 | 9 |
| 0 | 0 | 25 | 29 | 19 | 10 |
| 0 | 0 | 29 | 19 | 10 | 17 |
| 0 | 0 | 31 | 31 | 39 | 31 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,4,0,30,31,0,0,0,4,5,36,0,0,34,14,37,0,0,0,34,1,0,37],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,38,6,26,0,0,0,30,3,34,34,0,0,18,0,33,30,0,0,23,21,11,8],[7,34,0,0,0,0,6,0,0,0,0,0,0,0,6,35,17,13,0,0,11,0,21,34,0,0,3,37,5,11,0,0,1,37,30,30],[34,7,0,0,0,0,40,7,0,0,0,0,0,0,12,25,29,31,0,0,10,29,19,31,0,0,22,19,10,39,0,0,9,10,17,31] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | D10 | C4×D5 | 2+ (1+4) | 2- (1+4) | D4⋊8D10 | D4.10D10 |
| kernel | C42.91D10 | C4×Dic10 | C4×D20 | Dic5⋊4D4 | D5×C4⋊C4 | C4⋊C4⋊7D5 | C2×C4⋊Dic5 | C5×C42⋊C2 | C2×C4○D20 | C4○D20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 4 | 4 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{91}D_{10} % in TeX
G:=Group("C4^2.91D10"); // GroupNames label
G:=SmallGroup(320,1195);
// by ID
G=gap.SmallGroup(320,1195);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,570,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀