metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.14F5, Dic5⋊7M4(2), (C4×C20).6C4, (C4×D5).79D4, C4.10(C4⋊F5), C20.17(C4⋊C4), (C4×D5).21Q8, D10.26(C4⋊C4), Dic5⋊C8⋊4C2, C5⋊2(C4⋊M4(2)), (C4×Dic5).36C4, Dic5.10(C2×Q8), Dic5.28(C2×D4), (D5×C42).19C2, C10.7(C2×M4(2)), C2.8(D5⋊M4(2)), C22.63(C22×F5), (C4×Dic5).321C22, (C2×Dic5).318C23, C2.7(C2×C4⋊F5), C10.3(C2×C4⋊C4), (C2×C4×D5).29C4, (C2×C5⋊C8).2C22, (C2×C4).98(C2×F5), (C2×C4.F5).8C2, (C2×C20).122(C2×C4), (C2×C4×D5).391C22, (C2×C10).20(C22×C4), (C2×Dic5).168(C2×C4), (C22×D5).120(C2×C4), SmallGroup(320,1020)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 126 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×2], C10, C10 [×2], C42, C42 [×3], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4⋊M4(2), C4×Dic5, C4×Dic5 [×2], C4×C20, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], Dic5⋊C8 [×4], D5×C42, C2×C4.F5 [×2], C42.14F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2) [×2], C2×F5 [×3], C4⋊M4(2), C4⋊F5 [×2], C22×F5, D5⋊M4(2) [×2], C2×C4⋊F5, C42.14F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c3 >
►On 160 points
Generators in S160
(1 7 5 3)(2 4 6 8)(9 120 13 116)(10 117 14 113)(11 114 15 118)(12 119 16 115)(17 108 21 112)(18 105 22 109)(19 110 23 106)(20 107 24 111)(25 100 29 104)(26 97 30 101)(27 102 31 98)(28 99 32 103)(33 145 37 149)(34 150 38 146)(35 147 39 151)(36 152 40 148)(41 43 45 47)(42 48 46 44)(49 129 53 133)(50 134 54 130)(51 131 55 135)(52 136 56 132)(57 123 61 127)(58 128 62 124)(59 125 63 121)(60 122 64 126)(65 90 69 94)(66 95 70 91)(67 92 71 96)(68 89 72 93)(73 155 77 159)(74 160 78 156)(75 157 79 153)(76 154 80 158)(81 83 85 87)(82 88 86 84)(137 143 141 139)(138 140 142 144)
(1 82 141 44)(2 41 142 87)(3 84 143 46)(4 43 144 81)(5 86 137 48)(6 45 138 83)(7 88 139 42)(8 47 140 85)(9 122 76 21)(10 18 77 127)(11 124 78 23)(12 20 79 121)(13 126 80 17)(14 22 73 123)(15 128 74 19)(16 24 75 125)(25 134 65 149)(26 146 66 131)(27 136 67 151)(28 148 68 133)(29 130 69 145)(30 150 70 135)(31 132 71 147)(32 152 72 129)(33 100 54 90)(34 95 55 97)(35 102 56 92)(36 89 49 99)(37 104 50 94)(38 91 51 101)(39 98 52 96)(40 93 53 103)(57 117 105 159)(58 156 106 114)(59 119 107 153)(60 158 108 116)(61 113 109 155)(62 160 110 118)(63 115 111 157)(64 154 112 120)
(1 12 52 130 113)(2 131 13 114 53)(3 115 132 54 14)(4 55 116 15 133)(5 16 56 134 117)(6 135 9 118 49)(7 119 136 50 10)(8 51 120 11 129)(17 106 93 87 66)(18 88 107 67 94)(19 68 81 95 108)(20 96 69 109 82)(21 110 89 83 70)(22 84 111 71 90)(23 72 85 91 112)(24 92 65 105 86)(25 57 48 125 102)(26 126 58 103 41)(27 104 127 42 59)(28 43 97 60 128)(29 61 44 121 98)(30 122 62 99 45)(31 100 123 46 63)(32 47 101 64 124)(33 73 143 157 147)(34 158 74 148 144)(35 149 159 137 75)(36 138 150 76 160)(37 77 139 153 151)(38 154 78 152 140)(39 145 155 141 79)(40 142 146 80 156)
(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)(2,4,6,8)(9,120,13,116)(10,117,14,113)(11,114,15,118)(12,119,16,115)(17,108,21,112)(18,105,22,109)(19,110,23,106)(20,107,24,111)(25,100,29,104)(26,97,30,101)(27,102,31,98)(28,99,32,103)(33,145,37,149)(34,150,38,146)(35,147,39,151)(36,152,40,148)(41,43,45,47)(42,48,46,44)(49,129,53,133)(50,134,54,130)(51,131,55,135)(52,136,56,132)(57,123,61,127)(58,128,62,124)(59,125,63,121)(60,122,64,126)(65,90,69,94)(66,95,70,91)(67,92,71,96)(68,89,72,93)(73,155,77,159)(74,160,78,156)(75,157,79,153)(76,154,80,158)(81,83,85,87)(82,88,86,84)(137,143,141,139)(138,140,142,144), (1,82,141,44)(2,41,142,87)(3,84,143,46)(4,43,144,81)(5,86,137,48)(6,45,138,83)(7,88,139,42)(8,47,140,85)(9,122,76,21)(10,18,77,127)(11,124,78,23)(12,20,79,121)(13,126,80,17)(14,22,73,123)(15,128,74,19)(16,24,75,125)(25,134,65,149)(26,146,66,131)(27,136,67,151)(28,148,68,133)(29,130,69,145)(30,150,70,135)(31,132,71,147)(32,152,72,129)(33,100,54,90)(34,95,55,97)(35,102,56,92)(36,89,49,99)(37,104,50,94)(38,91,51,101)(39,98,52,96)(40,93,53,103)(57,117,105,159)(58,156,106,114)(59,119,107,153)(60,158,108,116)(61,113,109,155)(62,160,110,118)(63,115,111,157)(64,154,112,120), (1,12,52,130,113)(2,131,13,114,53)(3,115,132,54,14)(4,55,116,15,133)(5,16,56,134,117)(6,135,9,118,49)(7,119,136,50,10)(8,51,120,11,129)(17,106,93,87,66)(18,88,107,67,94)(19,68,81,95,108)(20,96,69,109,82)(21,110,89,83,70)(22,84,111,71,90)(23,72,85,91,112)(24,92,65,105,86)(25,57,48,125,102)(26,126,58,103,41)(27,104,127,42,59)(28,43,97,60,128)(29,61,44,121,98)(30,122,62,99,45)(31,100,123,46,63)(32,47,101,64,124)(33,73,143,157,147)(34,158,74,148,144)(35,149,159,137,75)(36,138,150,76,160)(37,77,139,153,151)(38,154,78,152,140)(39,145,155,141,79)(40,142,146,80,156), (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)(2,4,6,8)(9,120,13,116)(10,117,14,113)(11,114,15,118)(12,119,16,115)(17,108,21,112)(18,105,22,109)(19,110,23,106)(20,107,24,111)(25,100,29,104)(26,97,30,101)(27,102,31,98)(28,99,32,103)(33,145,37,149)(34,150,38,146)(35,147,39,151)(36,152,40,148)(41,43,45,47)(42,48,46,44)(49,129,53,133)(50,134,54,130)(51,131,55,135)(52,136,56,132)(57,123,61,127)(58,128,62,124)(59,125,63,121)(60,122,64,126)(65,90,69,94)(66,95,70,91)(67,92,71,96)(68,89,72,93)(73,155,77,159)(74,160,78,156)(75,157,79,153)(76,154,80,158)(81,83,85,87)(82,88,86,84)(137,143,141,139)(138,140,142,144), (1,82,141,44)(2,41,142,87)(3,84,143,46)(4,43,144,81)(5,86,137,48)(6,45,138,83)(7,88,139,42)(8,47,140,85)(9,122,76,21)(10,18,77,127)(11,124,78,23)(12,20,79,121)(13,126,80,17)(14,22,73,123)(15,128,74,19)(16,24,75,125)(25,134,65,149)(26,146,66,131)(27,136,67,151)(28,148,68,133)(29,130,69,145)(30,150,70,135)(31,132,71,147)(32,152,72,129)(33,100,54,90)(34,95,55,97)(35,102,56,92)(36,89,49,99)(37,104,50,94)(38,91,51,101)(39,98,52,96)(40,93,53,103)(57,117,105,159)(58,156,106,114)(59,119,107,153)(60,158,108,116)(61,113,109,155)(62,160,110,118)(63,115,111,157)(64,154,112,120), (1,12,52,130,113)(2,131,13,114,53)(3,115,132,54,14)(4,55,116,15,133)(5,16,56,134,117)(6,135,9,118,49)(7,119,136,50,10)(8,51,120,11,129)(17,106,93,87,66)(18,88,107,67,94)(19,68,81,95,108)(20,96,69,109,82)(21,110,89,83,70)(22,84,111,71,90)(23,72,85,91,112)(24,92,65,105,86)(25,57,48,125,102)(26,126,58,103,41)(27,104,127,42,59)(28,43,97,60,128)(29,61,44,121,98)(30,122,62,99,45)(31,100,123,46,63)(32,47,101,64,124)(33,73,143,157,147)(34,158,74,148,144)(35,149,159,137,75)(36,138,150,76,160)(37,77,139,153,151)(38,154,78,152,140)(39,145,155,141,79)(40,142,146,80,156), (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),(2,4,6,8),(9,120,13,116),(10,117,14,113),(11,114,15,118),(12,119,16,115),(17,108,21,112),(18,105,22,109),(19,110,23,106),(20,107,24,111),(25,100,29,104),(26,97,30,101),(27,102,31,98),(28,99,32,103),(33,145,37,149),(34,150,38,146),(35,147,39,151),(36,152,40,148),(41,43,45,47),(42,48,46,44),(49,129,53,133),(50,134,54,130),(51,131,55,135),(52,136,56,132),(57,123,61,127),(58,128,62,124),(59,125,63,121),(60,122,64,126),(65,90,69,94),(66,95,70,91),(67,92,71,96),(68,89,72,93),(73,155,77,159),(74,160,78,156),(75,157,79,153),(76,154,80,158),(81,83,85,87),(82,88,86,84),(137,143,141,139),(138,140,142,144)], [(1,82,141,44),(2,41,142,87),(3,84,143,46),(4,43,144,81),(5,86,137,48),(6,45,138,83),(7,88,139,42),(8,47,140,85),(9,122,76,21),(10,18,77,127),(11,124,78,23),(12,20,79,121),(13,126,80,17),(14,22,73,123),(15,128,74,19),(16,24,75,125),(25,134,65,149),(26,146,66,131),(27,136,67,151),(28,148,68,133),(29,130,69,145),(30,150,70,135),(31,132,71,147),(32,152,72,129),(33,100,54,90),(34,95,55,97),(35,102,56,92),(36,89,49,99),(37,104,50,94),(38,91,51,101),(39,98,52,96),(40,93,53,103),(57,117,105,159),(58,156,106,114),(59,119,107,153),(60,158,108,116),(61,113,109,155),(62,160,110,118),(63,115,111,157),(64,154,112,120)], [(1,12,52,130,113),(2,131,13,114,53),(3,115,132,54,14),(4,55,116,15,133),(5,16,56,134,117),(6,135,9,118,49),(7,119,136,50,10),(8,51,120,11,129),(17,106,93,87,66),(18,88,107,67,94),(19,68,81,95,108),(20,96,69,109,82),(21,110,89,83,70),(22,84,111,71,90),(23,72,85,91,112),(24,92,65,105,86),(25,57,48,125,102),(26,126,58,103,41),(27,104,127,42,59),(28,43,97,60,128),(29,61,44,121,98),(30,122,62,99,45),(31,100,123,46,63),(32,47,101,64,124),(33,73,143,157,147),(34,158,74,148,144),(35,149,159,137,75),(36,138,150,76,160),(37,77,139,153,151),(38,154,78,152,140),(39,145,155,141,79),(40,142,146,80,156)], [(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 |
| 23 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 14 | 14 |
| 0 | 0 | 27 | 34 | 27 | 0 |
| 0 | 0 | 0 | 27 | 34 | 27 |
| 0 | 0 | 14 | 14 | 0 | 7 |
| 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 |
| 10 | 31 | 0 | 0 | 0 | 0 |
| 5 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(41))| [32,23,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,27,0,14,0,0,0,34,27,14,0,0,14,27,34,0,0,0,14,0,27,7],[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],[10,5,0,0,0,0,31,31,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;
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 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | M4(2) | F5 | C2×F5 | C4⋊F5 | D5⋊M4(2) |
| kernel | C42.14F5 | Dic5⋊C8 | D5×C42 | C2×C4.F5 | C4×Dic5 | C4×C20 | C2×C4×D5 | C4×D5 | C4×D5 | Dic5 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 8 | 1 | 3 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{14}F_5 % in TeX
G:=Group("C4^2.14F5"); // GroupNames label
G:=SmallGroup(320,1020);
// by ID
G=gap.SmallGroup(320,1020);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,268,136,6278,1595]);
// 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^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀