metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.15F5, D10.9M4(2), C20.11M4(2), (C4×C20).7C4, C20⋊C8⋊8C2, C4.8(C4.F5), D10⋊C8.4C2, C5⋊1(C42.6C4), (C4×Dic5).27C4, (D5×C42).20C2, C10.8(C2×M4(2)), C10.C42⋊5C2, C2.9(D5⋊M4(2)), C10.3(C42⋊C2), Dic5.25(C4○D4), C22.64(C22×F5), (C2×Dic5).319C23, (C4×Dic5).322C22, C2.8(D10.C23), (C2×C4×D5).30C4, C2.7(C2×C4.F5), (C2×C5⋊C8).3C22, (C2×C4).99(C2×F5), (C2×C20).123(C2×C4), (C2×C4×D5).358C22, (C2×C10).21(C22×C4), (C2×Dic5).169(C2×C4), (C22×D5).121(C2×C4), SmallGroup(320,1021)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 110 conjugacy classes, 48 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×9], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C4×D5 [×6], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.6C4, C4×Dic5 [×3], C4×C20, C2×C5⋊C8 [×4], C2×C4×D5 [×3], C20⋊C8 [×2], C10.C42 [×2], D10⋊C8 [×2], D5×C42, C42.15F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×4], C22×C4, C4○D4 [×2], F5, C42⋊C2, C2×M4(2) [×2], C2×F5 [×3], C42.6C4, C4.F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), D10.C23, C42.15F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c3 >
►On 160 points
Generators in S160
(1 7 5 3)(2 64 6 60)(4 58 8 62)(9 154 13 158)(10 34 14 38)(11 156 15 160)(12 36 16 40)(17 91 21 95)(18 87 22 83)(19 93 23 89)(20 81 24 85)(25 107 29 111)(26 69 30 65)(27 109 31 105)(28 71 32 67)(33 76 37 80)(35 78 39 74)(41 147 45 151)(42 129 46 133)(43 149 47 145)(44 131 48 135)(49 101 53 97)(50 56 54 52)(51 103 55 99)(57 63 61 59)(66 127 70 123)(68 121 72 125)(73 155 77 159)(75 157 79 153)(82 119 86 115)(84 113 88 117)(90 114 94 118)(92 116 96 120)(98 104 102 100)(106 128 110 124)(108 122 112 126)(130 139 134 143)(132 141 136 137)(138 152 142 148)(140 146 144 150)
(1 98 57 52)(2 103 58 49)(3 100 59 54)(4 97 60 51)(5 102 61 56)(6 99 62 53)(7 104 63 50)(8 101 64 55)(9 30 80 122)(10 27 73 127)(11 32 74 124)(12 29 75 121)(13 26 76 126)(14 31 77 123)(15 28 78 128)(16 25 79 125)(17 135 119 146)(18 132 120 151)(19 129 113 148)(20 134 114 145)(21 131 115 150)(22 136 116 147)(23 133 117 152)(24 130 118 149)(33 112 154 65)(34 109 155 70)(35 106 156 67)(36 111 157 72)(37 108 158 69)(38 105 159 66)(39 110 160 71)(40 107 153 68)(41 87 141 92)(42 84 142 89)(43 81 143 94)(44 86 144 91)(45 83 137 96)(46 88 138 93)(47 85 139 90)(48 82 140 95)
(1 12 132 143 38)(2 144 13 39 133)(3 40 137 134 14)(4 135 33 15 138)(5 16 136 139 34)(6 140 9 35 129)(7 36 141 130 10)(8 131 37 11 142)(17 65 128 88 51)(18 81 66 52 121)(19 53 82 122 67)(20 123 54 68 83)(21 69 124 84 55)(22 85 70 56 125)(23 49 86 126 71)(24 127 50 72 87)(25 116 90 109 102)(26 110 117 103 91)(27 104 111 92 118)(28 93 97 119 112)(29 120 94 105 98)(30 106 113 99 95)(31 100 107 96 114)(32 89 101 115 108)(41 149 73 63 157)(42 64 150 158 74)(43 159 57 75 151)(44 76 160 152 58)(45 145 77 59 153)(46 60 146 154 78)(47 155 61 79 147)(48 80 156 148 62)
(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)
G:=sub<Sym(160)| (1,7,5,3)(2,64,6,60)(4,58,8,62)(9,154,13,158)(10,34,14,38)(11,156,15,160)(12,36,16,40)(17,91,21,95)(18,87,22,83)(19,93,23,89)(20,81,24,85)(25,107,29,111)(26,69,30,65)(27,109,31,105)(28,71,32,67)(33,76,37,80)(35,78,39,74)(41,147,45,151)(42,129,46,133)(43,149,47,145)(44,131,48,135)(49,101,53,97)(50,56,54,52)(51,103,55,99)(57,63,61,59)(66,127,70,123)(68,121,72,125)(73,155,77,159)(75,157,79,153)(82,119,86,115)(84,113,88,117)(90,114,94,118)(92,116,96,120)(98,104,102,100)(106,128,110,124)(108,122,112,126)(130,139,134,143)(132,141,136,137)(138,152,142,148)(140,146,144,150), (1,98,57,52)(2,103,58,49)(3,100,59,54)(4,97,60,51)(5,102,61,56)(6,99,62,53)(7,104,63,50)(8,101,64,55)(9,30,80,122)(10,27,73,127)(11,32,74,124)(12,29,75,121)(13,26,76,126)(14,31,77,123)(15,28,78,128)(16,25,79,125)(17,135,119,146)(18,132,120,151)(19,129,113,148)(20,134,114,145)(21,131,115,150)(22,136,116,147)(23,133,117,152)(24,130,118,149)(33,112,154,65)(34,109,155,70)(35,106,156,67)(36,111,157,72)(37,108,158,69)(38,105,159,66)(39,110,160,71)(40,107,153,68)(41,87,141,92)(42,84,142,89)(43,81,143,94)(44,86,144,91)(45,83,137,96)(46,88,138,93)(47,85,139,90)(48,82,140,95), (1,12,132,143,38)(2,144,13,39,133)(3,40,137,134,14)(4,135,33,15,138)(5,16,136,139,34)(6,140,9,35,129)(7,36,141,130,10)(8,131,37,11,142)(17,65,128,88,51)(18,81,66,52,121)(19,53,82,122,67)(20,123,54,68,83)(21,69,124,84,55)(22,85,70,56,125)(23,49,86,126,71)(24,127,50,72,87)(25,116,90,109,102)(26,110,117,103,91)(27,104,111,92,118)(28,93,97,119,112)(29,120,94,105,98)(30,106,113,99,95)(31,100,107,96,114)(32,89,101,115,108)(41,149,73,63,157)(42,64,150,158,74)(43,159,57,75,151)(44,76,160,152,58)(45,145,77,59,153)(46,60,146,154,78)(47,155,61,79,147)(48,80,156,148,62), (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)>;
G:=Group( (1,7,5,3)(2,64,6,60)(4,58,8,62)(9,154,13,158)(10,34,14,38)(11,156,15,160)(12,36,16,40)(17,91,21,95)(18,87,22,83)(19,93,23,89)(20,81,24,85)(25,107,29,111)(26,69,30,65)(27,109,31,105)(28,71,32,67)(33,76,37,80)(35,78,39,74)(41,147,45,151)(42,129,46,133)(43,149,47,145)(44,131,48,135)(49,101,53,97)(50,56,54,52)(51,103,55,99)(57,63,61,59)(66,127,70,123)(68,121,72,125)(73,155,77,159)(75,157,79,153)(82,119,86,115)(84,113,88,117)(90,114,94,118)(92,116,96,120)(98,104,102,100)(106,128,110,124)(108,122,112,126)(130,139,134,143)(132,141,136,137)(138,152,142,148)(140,146,144,150), (1,98,57,52)(2,103,58,49)(3,100,59,54)(4,97,60,51)(5,102,61,56)(6,99,62,53)(7,104,63,50)(8,101,64,55)(9,30,80,122)(10,27,73,127)(11,32,74,124)(12,29,75,121)(13,26,76,126)(14,31,77,123)(15,28,78,128)(16,25,79,125)(17,135,119,146)(18,132,120,151)(19,129,113,148)(20,134,114,145)(21,131,115,150)(22,136,116,147)(23,133,117,152)(24,130,118,149)(33,112,154,65)(34,109,155,70)(35,106,156,67)(36,111,157,72)(37,108,158,69)(38,105,159,66)(39,110,160,71)(40,107,153,68)(41,87,141,92)(42,84,142,89)(43,81,143,94)(44,86,144,91)(45,83,137,96)(46,88,138,93)(47,85,139,90)(48,82,140,95), (1,12,132,143,38)(2,144,13,39,133)(3,40,137,134,14)(4,135,33,15,138)(5,16,136,139,34)(6,140,9,35,129)(7,36,141,130,10)(8,131,37,11,142)(17,65,128,88,51)(18,81,66,52,121)(19,53,82,122,67)(20,123,54,68,83)(21,69,124,84,55)(22,85,70,56,125)(23,49,86,126,71)(24,127,50,72,87)(25,116,90,109,102)(26,110,117,103,91)(27,104,111,92,118)(28,93,97,119,112)(29,120,94,105,98)(30,106,113,99,95)(31,100,107,96,114)(32,89,101,115,108)(41,149,73,63,157)(42,64,150,158,74)(43,159,57,75,151)(44,76,160,152,58)(45,145,77,59,153)(46,60,146,154,78)(47,155,61,79,147)(48,80,156,148,62), (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) );
G=PermutationGroup([(1,7,5,3),(2,64,6,60),(4,58,8,62),(9,154,13,158),(10,34,14,38),(11,156,15,160),(12,36,16,40),(17,91,21,95),(18,87,22,83),(19,93,23,89),(20,81,24,85),(25,107,29,111),(26,69,30,65),(27,109,31,105),(28,71,32,67),(33,76,37,80),(35,78,39,74),(41,147,45,151),(42,129,46,133),(43,149,47,145),(44,131,48,135),(49,101,53,97),(50,56,54,52),(51,103,55,99),(57,63,61,59),(66,127,70,123),(68,121,72,125),(73,155,77,159),(75,157,79,153),(82,119,86,115),(84,113,88,117),(90,114,94,118),(92,116,96,120),(98,104,102,100),(106,128,110,124),(108,122,112,126),(130,139,134,143),(132,141,136,137),(138,152,142,148),(140,146,144,150)], [(1,98,57,52),(2,103,58,49),(3,100,59,54),(4,97,60,51),(5,102,61,56),(6,99,62,53),(7,104,63,50),(8,101,64,55),(9,30,80,122),(10,27,73,127),(11,32,74,124),(12,29,75,121),(13,26,76,126),(14,31,77,123),(15,28,78,128),(16,25,79,125),(17,135,119,146),(18,132,120,151),(19,129,113,148),(20,134,114,145),(21,131,115,150),(22,136,116,147),(23,133,117,152),(24,130,118,149),(33,112,154,65),(34,109,155,70),(35,106,156,67),(36,111,157,72),(37,108,158,69),(38,105,159,66),(39,110,160,71),(40,107,153,68),(41,87,141,92),(42,84,142,89),(43,81,143,94),(44,86,144,91),(45,83,137,96),(46,88,138,93),(47,85,139,90),(48,82,140,95)], [(1,12,132,143,38),(2,144,13,39,133),(3,40,137,134,14),(4,135,33,15,138),(5,16,136,139,34),(6,140,9,35,129),(7,36,141,130,10),(8,131,37,11,142),(17,65,128,88,51),(18,81,66,52,121),(19,53,82,122,67),(20,123,54,68,83),(21,69,124,84,55),(22,85,70,56,125),(23,49,86,126,71),(24,127,50,72,87),(25,116,90,109,102),(26,110,117,103,91),(27,104,111,92,118),(28,93,97,119,112),(29,120,94,105,98),(30,106,113,99,95),(31,100,107,96,114),(32,89,101,115,108),(41,149,73,63,157),(42,64,150,158,74),(43,159,57,75,151),(44,76,160,152,58),(45,145,77,59,153),(46,60,146,154,78),(47,155,61,79,147),(48,80,156,148,62)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 3 | 3 |
| 0 | 0 | 38 | 19 | 38 | 0 |
| 0 | 0 | 0 | 38 | 19 | 38 |
| 0 | 0 | 3 | 3 | 0 | 22 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 6 | 39 | 0 | 0 | 0 | 0 |
| 2 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 35 | 24 | 13 |
| 0 | 0 | 30 | 19 | 6 | 13 |
| 0 | 0 | 28 | 35 | 22 | 11 |
| 0 | 0 | 28 | 17 | 6 | 34 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[6,2,0,0,0,0,39,35,0,0,0,0,0,0,7,30,28,28,0,0,35,19,35,17,0,0,24,6,22,6,0,0,13,13,11,34] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | M4(2) | M4(2) | F5 | C2×F5 | C4.F5 | D5⋊M4(2) | D10.C23 |
| kernel | C42.15F5 | C20⋊C8 | C10.C42 | D10⋊C8 | D5×C42 | C4×Dic5 | C4×C20 | C2×C4×D5 | Dic5 | C20 | D10 | C42 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 3 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{15}F_5 % in TeX
G:=Group("C4^2.15F5"); // GroupNames label
G:=SmallGroup(320,1021);
// by ID
G=gap.SmallGroup(320,1021);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,232,758,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀