metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.5F5, C20.27C42, D10.9C42, D10.7M4(2), D5⋊C8⋊5C4, C4.20(C4×F5), (C4×C20).17C4, D5⋊1(C8⋊C4), C10.2(C2×C42), (C4×Dic5).44C4, (D5×C42).29C2, C10.1(C2×M4(2)), C2.1(D5⋊M4(2)), C10.C42⋊15C2, C22.25(C22×F5), Dic5.26(C22×C4), (C4×Dic5).318C22, (C2×Dic5).312C23, C5⋊C8⋊4(C2×C4), C5⋊1(C2×C8⋊C4), C2.5(C2×C4×F5), (C2×C4×D5).26C4, (C2×D5⋊C8).8C2, (C2×C4).95(C2×F5), (C4×D5).70(C2×C4), (C2×C5⋊C8).14C22, (C2×C20).166(C2×C4), (C2×C4×D5).409C22, (C2×C10).14(C22×C4), (C2×Dic5).162(C2×C4), (C22×D5).114(C2×C4), SmallGroup(320,1014)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 146 conjugacy classes, 76 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C2×C8 [×12], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C8⋊C4 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C8⋊C4, C4×Dic5, C4×Dic5 [×2], C4×C20, D5⋊C8 [×8], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], C10.C42 [×4], D5×C42, C2×D5⋊C8 [×2], C42.5F5
Quotients:
C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C2×F5 [×3], C2×C8⋊C4, C4×F5 [×2], C22×F5, D5⋊M4(2) [×2], C2×C4×F5, C42.5F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >
►On 160 points
Generators in S160
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 158 13 154)(10 159 14 155)(11 160 15 156)(12 153 16 157)(17 95 21 91)(18 96 22 92)(19 89 23 93)(20 90 24 94)(25 72 29 68)(26 65 30 69)(27 66 31 70)(28 67 32 71)(33 80 37 76)(34 73 38 77)(35 74 39 78)(36 75 40 79)(41 132 45 136)(42 133 46 129)(43 134 47 130)(44 135 48 131)(49 97 53 101)(50 98 54 102)(51 99 55 103)(52 100 56 104)(81 114 85 118)(82 115 86 119)(83 116 87 120)(84 117 88 113)(105 123 109 127)(106 124 110 128)(107 125 111 121)(108 126 112 122)(137 147 141 151)(138 148 142 152)(139 149 143 145)(140 150 144 146)
(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)| (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,158,13,154)(10,159,14,155)(11,160,15,156)(12,153,16,157)(17,95,21,91)(18,96,22,92)(19,89,23,93)(20,90,24,94)(25,72,29,68)(26,65,30,69)(27,66,31,70)(28,67,32,71)(33,80,37,76)(34,73,38,77)(35,74,39,78)(36,75,40,79)(41,132,45,136)(42,133,46,129)(43,134,47,130)(44,135,48,131)(49,97,53,101)(50,98,54,102)(51,99,55,103)(52,100,56,104)(81,114,85,118)(82,115,86,119)(83,116,87,120)(84,117,88,113)(105,123,109,127)(106,124,110,128)(107,125,111,121)(108,126,112,122)(137,147,141,151)(138,148,142,152)(139,149,143,145)(140,150,144,146), (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( (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,158,13,154)(10,159,14,155)(11,160,15,156)(12,153,16,157)(17,95,21,91)(18,96,22,92)(19,89,23,93)(20,90,24,94)(25,72,29,68)(26,65,30,69)(27,66,31,70)(28,67,32,71)(33,80,37,76)(34,73,38,77)(35,74,39,78)(36,75,40,79)(41,132,45,136)(42,133,46,129)(43,134,47,130)(44,135,48,131)(49,97,53,101)(50,98,54,102)(51,99,55,103)(52,100,56,104)(81,114,85,118)(82,115,86,119)(83,116,87,120)(84,117,88,113)(105,123,109,127)(106,124,110,128)(107,125,111,121)(108,126,112,122)(137,147,141,151)(138,148,142,152)(139,149,143,145)(140,150,144,146), (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([(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,158,13,154),(10,159,14,155),(11,160,15,156),(12,153,16,157),(17,95,21,91),(18,96,22,92),(19,89,23,93),(20,90,24,94),(25,72,29,68),(26,65,30,69),(27,66,31,70),(28,67,32,71),(33,80,37,76),(34,73,38,77),(35,74,39,78),(36,75,40,79),(41,132,45,136),(42,133,46,129),(43,134,47,130),(44,135,48,131),(49,97,53,101),(50,98,54,102),(51,99,55,103),(52,100,56,104),(81,114,85,118),(82,115,86,119),(83,116,87,120),(84,117,88,113),(105,123,109,127),(106,124,110,128),(107,125,111,121),(108,126,112,122),(137,147,141,151),(138,148,142,152),(139,149,143,145),(140,150,144,146)], [(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 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 18 | 2 | 0 | 0 | 0 | 0 |
| 22 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 0 | 38 | 38 |
| 0 | 0 | 3 | 22 | 3 | 0 |
| 0 | 0 | 0 | 3 | 22 | 3 |
| 0 | 0 | 38 | 38 | 0 | 19 |
| 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 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 19 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 26 | 26 |
| 0 | 0 | 26 | 26 | 0 | 17 |
| 0 | 0 | 15 | 32 | 15 | 0 |
| 0 | 0 | 24 | 9 | 9 | 24 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[18,22,0,0,0,0,2,23,0,0,0,0,0,0,19,3,0,38,0,0,0,22,3,38,0,0,38,3,22,0,0,0,38,0,3,19],[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],[3,19,0,0,0,0,12,38,0,0,0,0,0,0,17,26,15,24,0,0,0,26,32,9,0,0,26,0,15,9,0,0,26,17,0,24] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5 | 8A | ··· | 8P | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | F5 | C2×F5 | C4×F5 | D5⋊M4(2) |
| kernel | C42.5F5 | C10.C42 | D5×C42 | C2×D5⋊C8 | C4×Dic5 | C4×C20 | D5⋊C8 | C2×C4×D5 | D10 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 2 | 16 | 4 | 8 | 1 | 3 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._5F_5
% in TeX
G:=Group("C4^2.5F5"); // GroupNames label
G:=SmallGroup(320,1014);
// by ID
G=gap.SmallGroup(320,1014);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,100,136,6278,1595]);
// 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,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀