metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.155D10, C10.972- (1+4), C4⋊C4.211D10, C20⋊2Q8⋊33C2, C42.C2⋊11D5, (C2×C20).92C23, C4.Dic10⋊37C2, C42⋊D5.7C2, Dic5⋊3Q8⋊38C2, C20.131(C4○D4), (C4×C20).200C22, (C2×C10).241C24, D10⋊2Q8.13C2, C4.20(Q8⋊2D5), C4⋊Dic5.244C22, C22.262(C23×D5), C5⋊5(C22.35C24), (C2×Dic5).271C23, (C4×Dic5).154C22, (C22×D5).106C23, C2.60(D4.10D10), D10⋊C4.112C22, (C2×Dic10).188C22, C10.D4.124C22, C4⋊C4⋊D5.3C2, C10.118(C2×C4○D4), (C5×C42.C2)⋊14C2, C2.25(C2×Q8⋊2D5), (C2×C4×D5).140C22, (C5×C4⋊C4).196C22, (C2×C4).206(C22×D5), SmallGroup(320,1369)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 590 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, D5, C10, C10 [×2], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×14], C22×C4, C2×Q8 [×2], Dic5 [×7], C20 [×2], C20 [×6], D10 [×3], C2×C10, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C22×D5, C22.35C24, C4×Dic5, C4×Dic5 [×4], C10.D4 [×6], C4⋊Dic5 [×8], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×2], C2×C4×D5, C20⋊2Q8, C42⋊D5, Dic5⋊3Q8 [×2], C4.Dic10 [×4], D10⋊2Q8 [×2], C4⋊C4⋊D5 [×4], C5×C42.C2, C42.155D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- (1+4) [×2], C22×D5 [×7], C22.35C24, Q8⋊2D5 [×2], C23×D5, C2×Q8⋊2D5, D4.10D10 [×2], C42.155D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2b2c9 >
►On 160 points
Generators in S160
(1 94 50 68)(2 69 51 95)(3 96 52 70)(4 71 53 97)(5 98 54 72)(6 73 55 99)(7 100 56 74)(8 75 57 81)(9 82 58 76)(10 77 59 83)(11 84 60 78)(12 79 41 85)(13 86 42 80)(14 61 43 87)(15 88 44 62)(16 63 45 89)(17 90 46 64)(18 65 47 91)(19 92 48 66)(20 67 49 93)(21 124 149 114)(22 115 150 125)(23 126 151 116)(24 117 152 127)(25 128 153 118)(26 119 154 129)(27 130 155 120)(28 101 156 131)(29 132 157 102)(30 103 158 133)(31 134 159 104)(32 105 160 135)(33 136 141 106)(34 107 142 137)(35 138 143 108)(36 109 144 139)(37 140 145 110)(38 111 146 121)(39 122 147 112)(40 113 148 123)
(1 142 11 152)(2 25 12 35)(3 144 13 154)(4 27 14 37)(5 146 15 156)(6 29 16 39)(7 148 17 158)(8 31 18 21)(9 150 19 160)(10 33 20 23)(22 48 32 58)(24 50 34 60)(26 52 36 42)(28 54 38 44)(30 56 40 46)(41 143 51 153)(43 145 53 155)(45 147 55 157)(47 149 57 159)(49 151 59 141)(61 140 71 130)(62 101 72 111)(63 122 73 132)(64 103 74 113)(65 124 75 134)(66 105 76 115)(67 126 77 136)(68 107 78 117)(69 128 79 138)(70 109 80 119)(81 104 91 114)(82 125 92 135)(83 106 93 116)(84 127 94 137)(85 108 95 118)(86 129 96 139)(87 110 97 120)(88 131 98 121)(89 112 99 102)(90 133 100 123)
(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 50 59)(2 58 51 9)(3 8 52 57)(4 56 53 7)(5 6 54 55)(11 20 60 49)(12 48 41 19)(13 18 42 47)(14 46 43 17)(15 16 44 45)(21 36 149 144)(22 143 150 35)(23 34 151 142)(24 141 152 33)(25 32 153 160)(26 159 154 31)(27 30 155 158)(28 157 156 29)(37 40 145 148)(38 147 146 39)(61 90 87 64)(62 63 88 89)(65 86 91 80)(66 79 92 85)(67 84 93 78)(68 77 94 83)(69 82 95 76)(70 75 96 81)(71 100 97 74)(72 73 98 99)(101 132 131 102)(103 130 133 120)(104 119 134 129)(105 128 135 118)(106 117 136 127)(107 126 137 116)(108 115 138 125)(109 124 139 114)(110 113 140 123)(111 122 121 112)
G:=sub<Sym(160)| (1,94,50,68)(2,69,51,95)(3,96,52,70)(4,71,53,97)(5,98,54,72)(6,73,55,99)(7,100,56,74)(8,75,57,81)(9,82,58,76)(10,77,59,83)(11,84,60,78)(12,79,41,85)(13,86,42,80)(14,61,43,87)(15,88,44,62)(16,63,45,89)(17,90,46,64)(18,65,47,91)(19,92,48,66)(20,67,49,93)(21,124,149,114)(22,115,150,125)(23,126,151,116)(24,117,152,127)(25,128,153,118)(26,119,154,129)(27,130,155,120)(28,101,156,131)(29,132,157,102)(30,103,158,133)(31,134,159,104)(32,105,160,135)(33,136,141,106)(34,107,142,137)(35,138,143,108)(36,109,144,139)(37,140,145,110)(38,111,146,121)(39,122,147,112)(40,113,148,123), (1,142,11,152)(2,25,12,35)(3,144,13,154)(4,27,14,37)(5,146,15,156)(6,29,16,39)(7,148,17,158)(8,31,18,21)(9,150,19,160)(10,33,20,23)(22,48,32,58)(24,50,34,60)(26,52,36,42)(28,54,38,44)(30,56,40,46)(41,143,51,153)(43,145,53,155)(45,147,55,157)(47,149,57,159)(49,151,59,141)(61,140,71,130)(62,101,72,111)(63,122,73,132)(64,103,74,113)(65,124,75,134)(66,105,76,115)(67,126,77,136)(68,107,78,117)(69,128,79,138)(70,109,80,119)(81,104,91,114)(82,125,92,135)(83,106,93,116)(84,127,94,137)(85,108,95,118)(86,129,96,139)(87,110,97,120)(88,131,98,121)(89,112,99,102)(90,133,100,123), (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), (1,10,50,59)(2,58,51,9)(3,8,52,57)(4,56,53,7)(5,6,54,55)(11,20,60,49)(12,48,41,19)(13,18,42,47)(14,46,43,17)(15,16,44,45)(21,36,149,144)(22,143,150,35)(23,34,151,142)(24,141,152,33)(25,32,153,160)(26,159,154,31)(27,30,155,158)(28,157,156,29)(37,40,145,148)(38,147,146,39)(61,90,87,64)(62,63,88,89)(65,86,91,80)(66,79,92,85)(67,84,93,78)(68,77,94,83)(69,82,95,76)(70,75,96,81)(71,100,97,74)(72,73,98,99)(101,132,131,102)(103,130,133,120)(104,119,134,129)(105,128,135,118)(106,117,136,127)(107,126,137,116)(108,115,138,125)(109,124,139,114)(110,113,140,123)(111,122,121,112)>;
G:=Group( (1,94,50,68)(2,69,51,95)(3,96,52,70)(4,71,53,97)(5,98,54,72)(6,73,55,99)(7,100,56,74)(8,75,57,81)(9,82,58,76)(10,77,59,83)(11,84,60,78)(12,79,41,85)(13,86,42,80)(14,61,43,87)(15,88,44,62)(16,63,45,89)(17,90,46,64)(18,65,47,91)(19,92,48,66)(20,67,49,93)(21,124,149,114)(22,115,150,125)(23,126,151,116)(24,117,152,127)(25,128,153,118)(26,119,154,129)(27,130,155,120)(28,101,156,131)(29,132,157,102)(30,103,158,133)(31,134,159,104)(32,105,160,135)(33,136,141,106)(34,107,142,137)(35,138,143,108)(36,109,144,139)(37,140,145,110)(38,111,146,121)(39,122,147,112)(40,113,148,123), (1,142,11,152)(2,25,12,35)(3,144,13,154)(4,27,14,37)(5,146,15,156)(6,29,16,39)(7,148,17,158)(8,31,18,21)(9,150,19,160)(10,33,20,23)(22,48,32,58)(24,50,34,60)(26,52,36,42)(28,54,38,44)(30,56,40,46)(41,143,51,153)(43,145,53,155)(45,147,55,157)(47,149,57,159)(49,151,59,141)(61,140,71,130)(62,101,72,111)(63,122,73,132)(64,103,74,113)(65,124,75,134)(66,105,76,115)(67,126,77,136)(68,107,78,117)(69,128,79,138)(70,109,80,119)(81,104,91,114)(82,125,92,135)(83,106,93,116)(84,127,94,137)(85,108,95,118)(86,129,96,139)(87,110,97,120)(88,131,98,121)(89,112,99,102)(90,133,100,123), (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), (1,10,50,59)(2,58,51,9)(3,8,52,57)(4,56,53,7)(5,6,54,55)(11,20,60,49)(12,48,41,19)(13,18,42,47)(14,46,43,17)(15,16,44,45)(21,36,149,144)(22,143,150,35)(23,34,151,142)(24,141,152,33)(25,32,153,160)(26,159,154,31)(27,30,155,158)(28,157,156,29)(37,40,145,148)(38,147,146,39)(61,90,87,64)(62,63,88,89)(65,86,91,80)(66,79,92,85)(67,84,93,78)(68,77,94,83)(69,82,95,76)(70,75,96,81)(71,100,97,74)(72,73,98,99)(101,132,131,102)(103,130,133,120)(104,119,134,129)(105,128,135,118)(106,117,136,127)(107,126,137,116)(108,115,138,125)(109,124,139,114)(110,113,140,123)(111,122,121,112) );
G=PermutationGroup([(1,94,50,68),(2,69,51,95),(3,96,52,70),(4,71,53,97),(5,98,54,72),(6,73,55,99),(7,100,56,74),(8,75,57,81),(9,82,58,76),(10,77,59,83),(11,84,60,78),(12,79,41,85),(13,86,42,80),(14,61,43,87),(15,88,44,62),(16,63,45,89),(17,90,46,64),(18,65,47,91),(19,92,48,66),(20,67,49,93),(21,124,149,114),(22,115,150,125),(23,126,151,116),(24,117,152,127),(25,128,153,118),(26,119,154,129),(27,130,155,120),(28,101,156,131),(29,132,157,102),(30,103,158,133),(31,134,159,104),(32,105,160,135),(33,136,141,106),(34,107,142,137),(35,138,143,108),(36,109,144,139),(37,140,145,110),(38,111,146,121),(39,122,147,112),(40,113,148,123)], [(1,142,11,152),(2,25,12,35),(3,144,13,154),(4,27,14,37),(5,146,15,156),(6,29,16,39),(7,148,17,158),(8,31,18,21),(9,150,19,160),(10,33,20,23),(22,48,32,58),(24,50,34,60),(26,52,36,42),(28,54,38,44),(30,56,40,46),(41,143,51,153),(43,145,53,155),(45,147,55,157),(47,149,57,159),(49,151,59,141),(61,140,71,130),(62,101,72,111),(63,122,73,132),(64,103,74,113),(65,124,75,134),(66,105,76,115),(67,126,77,136),(68,107,78,117),(69,128,79,138),(70,109,80,119),(81,104,91,114),(82,125,92,135),(83,106,93,116),(84,127,94,137),(85,108,95,118),(86,129,96,139),(87,110,97,120),(88,131,98,121),(89,112,99,102),(90,133,100,123)], [(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)], [(1,10,50,59),(2,58,51,9),(3,8,52,57),(4,56,53,7),(5,6,54,55),(11,20,60,49),(12,48,41,19),(13,18,42,47),(14,46,43,17),(15,16,44,45),(21,36,149,144),(22,143,150,35),(23,34,151,142),(24,141,152,33),(25,32,153,160),(26,159,154,31),(27,30,155,158),(28,157,156,29),(37,40,145,148),(38,147,146,39),(61,90,87,64),(62,63,88,89),(65,86,91,80),(66,79,92,85),(67,84,93,78),(68,77,94,83),(69,82,95,76),(70,75,96,81),(71,100,97,74),(72,73,98,99),(101,132,131,102),(103,130,133,120),(104,119,134,129),(105,128,135,118),(106,117,136,127),(107,126,137,116),(108,115,138,125),(109,124,139,114),(110,113,140,123),(111,122,121,112)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 30 |
| 0 | 0 | 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 22 | 19 | 32 | 9 | 0 | 0 | 0 | 0 |
| 22 | 12 | 32 | 10 | 0 | 0 | 0 | 0 |
| 32 | 9 | 19 | 22 | 0 | 0 | 0 | 0 |
| 32 | 10 | 19 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 8 | 21 | 18 |
| 0 | 0 | 0 | 0 | 18 | 25 | 20 | 0 |
| 0 | 0 | 0 | 0 | 20 | 23 | 34 | 33 |
| 0 | 0 | 0 | 0 | 21 | 0 | 23 | 16 |
| 19 | 22 | 9 | 32 | 0 | 0 | 0 | 0 |
| 12 | 22 | 10 | 32 | 0 | 0 | 0 | 0 |
| 9 | 32 | 22 | 19 | 0 | 0 | 0 | 0 |
| 10 | 32 | 29 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 | 21 | 3 |
| 0 | 0 | 0 | 0 | 23 | 7 | 20 | 20 |
| 0 | 0 | 0 | 0 | 21 | 3 | 34 | 1 |
| 0 | 0 | 0 | 0 | 20 | 20 | 23 | 7 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[22,22,32,32,0,0,0,0,19,12,9,10,0,0,0,0,32,32,19,19,0,0,0,0,9,10,22,29,0,0,0,0,0,0,0,0,7,18,20,21,0,0,0,0,8,25,23,0,0,0,0,0,21,20,34,23,0,0,0,0,18,0,33,16],[19,12,9,10,0,0,0,0,22,22,32,32,0,0,0,0,9,10,22,29,0,0,0,0,32,32,19,19,0,0,0,0,0,0,0,0,34,23,21,20,0,0,0,0,1,7,3,20,0,0,0,0,21,20,34,23,0,0,0,0,3,20,1,7] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2- (1+4) | Q8⋊2D5 | D4.10D10 |
| kernel | C42.155D10 | C20⋊2Q8 | C42⋊D5 | Dic5⋊3Q8 | C4.Dic10 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{155}D_{10} % in TeX
G:=Group("C4^2.155D10"); // GroupNames label
G:=SmallGroup(320,1369);
// by ID
G=gap.SmallGroup(320,1369);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀