metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.174D10, C10.372- (1+4), C10.822+ (1+4), C4⋊Q8⋊12D5, (C4×D5)⋊2Q8, C20⋊Q8⋊45C2, C4.41(Q8×D5), D10.6(C2×Q8), C20.55(C2×Q8), C4⋊C4.124D10, (C2×Q8).86D10, Dic5.7(C2×Q8), D10⋊Q8.4C2, C20.6Q8⋊24C2, C4.Dic10⋊43C2, C42⋊D5.8C2, Dic5⋊Q8⋊27C2, C10.49(C22×Q8), (C2×C20).106C23, (C4×C20).214C22, (C2×C10).273C24, D10⋊3Q8.13C2, C2.86(D4⋊6D10), Dic5.Q8⋊41C2, C4⋊Dic5.252C22, (Q8×C10).140C22, C22.294(C23×D5), D10⋊C4.52C22, C5⋊5(C23.41C23), (C2×Dic5).144C23, (C4×Dic5).170C22, (C22×D5).244C23, C2.38(Q8.10D10), (C2×Dic10).197C22, C10.D4.167C22, C2.32(C2×Q8×D5), (C5×C4⋊Q8)⋊15C2, (D5×C4⋊C4).13C2, C4⋊C4⋊7D5.15C2, (C2×C4×D5).155C22, (C5×C4⋊C4).216C22, (C2×C4).219(C22×D5), SmallGroup(320,1401)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 654 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×16], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8, C4⋊Q8 [×3], Dic10 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C23.41C23, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×10], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×C10 [×2], C20.6Q8, C42⋊D5, C20⋊Q8, Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, D10⋊Q8 [×2], Dic5⋊Q8 [×2], D10⋊3Q8 [×2], C5×C4⋊Q8, C42.174D10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ (1+4), 2- (1+4), C22×D5 [×7], C23.41C23, Q8×D5 [×2], C23×D5, D4⋊6D10, C2×Q8×D5, Q8.10D10, C42.174D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 117 50 24)(2 25 51 118)(3 119 52 26)(4 27 53 120)(5 101 54 28)(6 29 55 102)(7 103 56 30)(8 31 57 104)(9 105 58 32)(10 33 59 106)(11 107 60 34)(12 35 41 108)(13 109 42 36)(14 37 43 110)(15 111 44 38)(16 39 45 112)(17 113 46 40)(18 21 47 114)(19 115 48 22)(20 23 49 116)(61 88 148 133)(62 134 149 89)(63 90 150 135)(64 136 151 91)(65 92 152 137)(66 138 153 93)(67 94 154 139)(68 140 155 95)(69 96 156 121)(70 122 157 97)(71 98 158 123)(72 124 159 99)(73 100 160 125)(74 126 141 81)(75 82 142 127)(76 128 143 83)(77 84 144 129)(78 130 145 85)(79 86 146 131)(80 132 147 87)
(1 155 60 78)(2 79 41 156)(3 157 42 80)(4 61 43 158)(5 159 44 62)(6 63 45 160)(7 141 46 64)(8 65 47 142)(9 143 48 66)(10 67 49 144)(11 145 50 68)(12 69 51 146)(13 147 52 70)(14 71 53 148)(15 149 54 72)(16 73 55 150)(17 151 56 74)(18 75 57 152)(19 153 58 76)(20 77 59 154)(21 82 104 137)(22 138 105 83)(23 84 106 139)(24 140 107 85)(25 86 108 121)(26 122 109 87)(27 88 110 123)(28 124 111 89)(29 90 112 125)(30 126 113 91)(31 92 114 127)(32 128 115 93)(33 94 116 129)(34 130 117 95)(35 96 118 131)(36 132 119 97)(37 98 120 133)(38 134 101 99)(39 100 102 135)(40 136 103 81)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 58 51 48)(42 47 52 57)(43 56 53 46)(44 45 54 55)(49 50 59 60)(61 141 71 151)(62 150 72 160)(63 159 73 149)(64 148 74 158)(65 157 75 147)(66 146 76 156)(67 155 77 145)(68 144 78 154)(69 153 79 143)(70 142 80 152)(81 123 91 133)(82 132 92 122)(83 121 93 131)(84 130 94 140)(85 139 95 129)(86 128 96 138)(87 137 97 127)(88 126 98 136)(89 135 99 125)(90 124 100 134)(101 102 111 112)(103 120 113 110)(104 109 114 119)(105 118 115 108)(106 107 116 117)
G:=sub<Sym(160)| (1,117,50,24)(2,25,51,118)(3,119,52,26)(4,27,53,120)(5,101,54,28)(6,29,55,102)(7,103,56,30)(8,31,57,104)(9,105,58,32)(10,33,59,106)(11,107,60,34)(12,35,41,108)(13,109,42,36)(14,37,43,110)(15,111,44,38)(16,39,45,112)(17,113,46,40)(18,21,47,114)(19,115,48,22)(20,23,49,116)(61,88,148,133)(62,134,149,89)(63,90,150,135)(64,136,151,91)(65,92,152,137)(66,138,153,93)(67,94,154,139)(68,140,155,95)(69,96,156,121)(70,122,157,97)(71,98,158,123)(72,124,159,99)(73,100,160,125)(74,126,141,81)(75,82,142,127)(76,128,143,83)(77,84,144,129)(78,130,145,85)(79,86,146,131)(80,132,147,87), (1,155,60,78)(2,79,41,156)(3,157,42,80)(4,61,43,158)(5,159,44,62)(6,63,45,160)(7,141,46,64)(8,65,47,142)(9,143,48,66)(10,67,49,144)(11,145,50,68)(12,69,51,146)(13,147,52,70)(14,71,53,148)(15,149,54,72)(16,73,55,150)(17,151,56,74)(18,75,57,152)(19,153,58,76)(20,77,59,154)(21,82,104,137)(22,138,105,83)(23,84,106,139)(24,140,107,85)(25,86,108,121)(26,122,109,87)(27,88,110,123)(28,124,111,89)(29,90,112,125)(30,126,113,91)(31,92,114,127)(32,128,115,93)(33,94,116,129)(34,130,117,95)(35,96,118,131)(36,132,119,97)(37,98,120,133)(38,134,101,99)(39,100,102,135)(40,136,103,81), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,58,51,48)(42,47,52,57)(43,56,53,46)(44,45,54,55)(49,50,59,60)(61,141,71,151)(62,150,72,160)(63,159,73,149)(64,148,74,158)(65,157,75,147)(66,146,76,156)(67,155,77,145)(68,144,78,154)(69,153,79,143)(70,142,80,152)(81,123,91,133)(82,132,92,122)(83,121,93,131)(84,130,94,140)(85,139,95,129)(86,128,96,138)(87,137,97,127)(88,126,98,136)(89,135,99,125)(90,124,100,134)(101,102,111,112)(103,120,113,110)(104,109,114,119)(105,118,115,108)(106,107,116,117)>;
G:=Group( (1,117,50,24)(2,25,51,118)(3,119,52,26)(4,27,53,120)(5,101,54,28)(6,29,55,102)(7,103,56,30)(8,31,57,104)(9,105,58,32)(10,33,59,106)(11,107,60,34)(12,35,41,108)(13,109,42,36)(14,37,43,110)(15,111,44,38)(16,39,45,112)(17,113,46,40)(18,21,47,114)(19,115,48,22)(20,23,49,116)(61,88,148,133)(62,134,149,89)(63,90,150,135)(64,136,151,91)(65,92,152,137)(66,138,153,93)(67,94,154,139)(68,140,155,95)(69,96,156,121)(70,122,157,97)(71,98,158,123)(72,124,159,99)(73,100,160,125)(74,126,141,81)(75,82,142,127)(76,128,143,83)(77,84,144,129)(78,130,145,85)(79,86,146,131)(80,132,147,87), (1,155,60,78)(2,79,41,156)(3,157,42,80)(4,61,43,158)(5,159,44,62)(6,63,45,160)(7,141,46,64)(8,65,47,142)(9,143,48,66)(10,67,49,144)(11,145,50,68)(12,69,51,146)(13,147,52,70)(14,71,53,148)(15,149,54,72)(16,73,55,150)(17,151,56,74)(18,75,57,152)(19,153,58,76)(20,77,59,154)(21,82,104,137)(22,138,105,83)(23,84,106,139)(24,140,107,85)(25,86,108,121)(26,122,109,87)(27,88,110,123)(28,124,111,89)(29,90,112,125)(30,126,113,91)(31,92,114,127)(32,128,115,93)(33,94,116,129)(34,130,117,95)(35,96,118,131)(36,132,119,97)(37,98,120,133)(38,134,101,99)(39,100,102,135)(40,136,103,81), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,58,51,48)(42,47,52,57)(43,56,53,46)(44,45,54,55)(49,50,59,60)(61,141,71,151)(62,150,72,160)(63,159,73,149)(64,148,74,158)(65,157,75,147)(66,146,76,156)(67,155,77,145)(68,144,78,154)(69,153,79,143)(70,142,80,152)(81,123,91,133)(82,132,92,122)(83,121,93,131)(84,130,94,140)(85,139,95,129)(86,128,96,138)(87,137,97,127)(88,126,98,136)(89,135,99,125)(90,124,100,134)(101,102,111,112)(103,120,113,110)(104,109,114,119)(105,118,115,108)(106,107,116,117) );
G=PermutationGroup([(1,117,50,24),(2,25,51,118),(3,119,52,26),(4,27,53,120),(5,101,54,28),(6,29,55,102),(7,103,56,30),(8,31,57,104),(9,105,58,32),(10,33,59,106),(11,107,60,34),(12,35,41,108),(13,109,42,36),(14,37,43,110),(15,111,44,38),(16,39,45,112),(17,113,46,40),(18,21,47,114),(19,115,48,22),(20,23,49,116),(61,88,148,133),(62,134,149,89),(63,90,150,135),(64,136,151,91),(65,92,152,137),(66,138,153,93),(67,94,154,139),(68,140,155,95),(69,96,156,121),(70,122,157,97),(71,98,158,123),(72,124,159,99),(73,100,160,125),(74,126,141,81),(75,82,142,127),(76,128,143,83),(77,84,144,129),(78,130,145,85),(79,86,146,131),(80,132,147,87)], [(1,155,60,78),(2,79,41,156),(3,157,42,80),(4,61,43,158),(5,159,44,62),(6,63,45,160),(7,141,46,64),(8,65,47,142),(9,143,48,66),(10,67,49,144),(11,145,50,68),(12,69,51,146),(13,147,52,70),(14,71,53,148),(15,149,54,72),(16,73,55,150),(17,151,56,74),(18,75,57,152),(19,153,58,76),(20,77,59,154),(21,82,104,137),(22,138,105,83),(23,84,106,139),(24,140,107,85),(25,86,108,121),(26,122,109,87),(27,88,110,123),(28,124,111,89),(29,90,112,125),(30,126,113,91),(31,92,114,127),(32,128,115,93),(33,94,116,129),(34,130,117,95),(35,96,118,131),(36,132,119,97),(37,98,120,133),(38,134,101,99),(39,100,102,135),(40,136,103,81)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,58,51,48),(42,47,52,57),(43,56,53,46),(44,45,54,55),(49,50,59,60),(61,141,71,151),(62,150,72,160),(63,159,73,149),(64,148,74,158),(65,157,75,147),(66,146,76,156),(67,155,77,145),(68,144,78,154),(69,153,79,143),(70,142,80,152),(81,123,91,133),(82,132,92,122),(83,121,93,131),(84,130,94,140),(85,139,95,129),(86,128,96,138),(87,137,97,127),(88,126,98,136),(89,135,99,125),(90,124,100,134),(101,102,111,112),(103,120,113,110),(104,109,114,119),(105,118,115,108),(106,107,116,117)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 27 | 7 | 0 | 0 | 0 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 27 | 27 |
| 0 | 0 | 7 | 1 | 14 | 2 |
| 0 | 0 | 7 | 7 | 7 | 7 |
| 0 | 0 | 34 | 40 | 34 | 40 |
| 14 | 34 | 0 | 0 | 0 | 0 |
| 34 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 27 | 27 |
| 0 | 0 | 1 | 7 | 2 | 14 |
| 0 | 0 | 7 | 7 | 7 | 7 |
| 0 | 0 | 40 | 34 | 40 | 34 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,11,0,0,0,0,0,0,11,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[27,7,0,0,0,0,7,14,0,0,0,0,0,0,34,7,7,34,0,0,34,1,7,40,0,0,27,14,7,34,0,0,27,2,7,40],[14,34,0,0,0,0,34,27,0,0,0,0,0,0,34,1,7,40,0,0,34,7,7,34,0,0,27,2,7,40,0,0,27,14,7,34] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | Q8×D5 | D4⋊6D10 | Q8.10D10 |
| kernel | C42.174D10 | C20.6Q8 | C42⋊D5 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10⋊Q8 | Dic5⋊Q8 | D10⋊3Q8 | C5×C4⋊Q8 | C4×D5 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{174}D_{10} % in TeX
G:=Group("C4^2.174D10"); // GroupNames label
G:=SmallGroup(320,1401);
// by ID
G=gap.SmallGroup(320,1401);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,136,12550]);
// 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,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀