metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.6F5, Dic5.10M4(2), (C4×D5)⋊7C8, C20.22(C2×C8), (C4×C20).18C4, C4.17(D5⋊C8), D10.13(C2×C8), C10.2(C22×C8), D10⋊C8.6C2, Dic5.14(C2×C8), (C4×Dic5).25C4, (D5×C42).30C2, C10.3(C2×M4(2)), Dic5⋊C8⋊14C2, C5⋊1(C42.12C4), C2.3(D5⋊M4(2)), C10.1(C42⋊C2), Dic5.23(C4○D4), C22.27(C22×F5), (C2×Dic5).314C23, (C4×Dic5).354C22, C2.1(D10.C23), (C4×C5⋊C8)⋊8C2, C2.4(C2×D5⋊C8), (C2×C4×D5).27C4, (C2×C4).96(C2×F5), (C2×C5⋊C8).16C22, (C2×C20).167(C2×C4), (C2×C4×D5).356C22, (C2×C10).16(C22×C4), (C2×Dic5).164(C2×C4), (C22×D5).116(C2×C4), SmallGroup(320,1016)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 118 conjugacy classes, 56 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], C22×C4 [×3], Dic5 [×6], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.12C4, C4×Dic5 [×3], C4×C20, C2×C5⋊C8 [×4], C2×C4×D5 [×3], C4×C5⋊C8 [×2], D10⋊C8 [×2], Dic5⋊C8 [×2], D5×C42, C42.6F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5 [×3], C42.12C4, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D5⋊M4(2), D10.C23, C42.6F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >
►On 160 points
Generators in S160
(1 83 57 71)(2 84 58 72)(3 85 59 65)(4 86 60 66)(5 87 61 67)(6 88 62 68)(7 81 63 69)(8 82 64 70)(9 89 24 130)(10 90 17 131)(11 91 18 132)(12 92 19 133)(13 93 20 134)(14 94 21 135)(15 95 22 136)(16 96 23 129)(25 79 125 151)(26 80 126 152)(27 73 127 145)(28 74 128 146)(29 75 121 147)(30 76 122 148)(31 77 123 149)(32 78 124 150)(33 43 104 143)(34 44 97 144)(35 45 98 137)(36 46 99 138)(37 47 100 139)(38 48 101 140)(39 41 102 141)(40 42 103 142)(49 114 106 160)(50 115 107 153)(51 116 108 154)(52 117 109 155)(53 118 110 156)(54 119 111 157)(55 120 112 158)(56 113 105 159)
(1 7 5 3)(2 64 6 60)(4 58 8 62)(9 37 13 33)(10 101 14 97)(11 39 15 35)(12 103 16 99)(17 38 21 34)(18 102 22 98)(19 40 23 36)(20 104 24 100)(25 107 29 111)(26 51 30 55)(27 109 31 105)(28 53 32 49)(41 95 45 91)(42 129 46 133)(43 89 47 93)(44 131 48 135)(50 121 54 125)(52 123 56 127)(57 63 61 59)(65 71 69 67)(66 84 70 88)(68 86 72 82)(73 155 77 159)(74 118 78 114)(75 157 79 153)(76 120 80 116)(81 87 85 83)(90 140 94 144)(92 142 96 138)(106 128 110 124)(108 122 112 126)(113 145 117 149)(115 147 119 151)(130 139 134 143)(132 141 136 137)(146 156 150 160)(148 158 152 154)
(1 147 132 143 113)(2 144 148 114 133)(3 115 137 134 149)(4 135 116 150 138)(5 151 136 139 117)(6 140 152 118 129)(7 119 141 130 145)(8 131 120 146 142)(9 27 81 111 39)(10 112 28 40 82)(11 33 105 83 29)(12 84 34 30 106)(13 31 85 107 35)(14 108 32 36 86)(15 37 109 87 25)(16 88 38 26 110)(17 55 128 103 70)(18 104 56 71 121)(19 72 97 122 49)(20 123 65 50 98)(21 51 124 99 66)(22 100 52 67 125)(23 68 101 126 53)(24 127 69 54 102)(41 89 73 63 157)(42 64 90 158 74)(43 159 57 75 91)(44 76 160 92 58)(45 93 77 59 153)(46 60 94 154 78)(47 155 61 79 95)(48 80 156 96 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)| (1,83,57,71)(2,84,58,72)(3,85,59,65)(4,86,60,66)(5,87,61,67)(6,88,62,68)(7,81,63,69)(8,82,64,70)(9,89,24,130)(10,90,17,131)(11,91,18,132)(12,92,19,133)(13,93,20,134)(14,94,21,135)(15,95,22,136)(16,96,23,129)(25,79,125,151)(26,80,126,152)(27,73,127,145)(28,74,128,146)(29,75,121,147)(30,76,122,148)(31,77,123,149)(32,78,124,150)(33,43,104,143)(34,44,97,144)(35,45,98,137)(36,46,99,138)(37,47,100,139)(38,48,101,140)(39,41,102,141)(40,42,103,142)(49,114,106,160)(50,115,107,153)(51,116,108,154)(52,117,109,155)(53,118,110,156)(54,119,111,157)(55,120,112,158)(56,113,105,159), (1,7,5,3)(2,64,6,60)(4,58,8,62)(9,37,13,33)(10,101,14,97)(11,39,15,35)(12,103,16,99)(17,38,21,34)(18,102,22,98)(19,40,23,36)(20,104,24,100)(25,107,29,111)(26,51,30,55)(27,109,31,105)(28,53,32,49)(41,95,45,91)(42,129,46,133)(43,89,47,93)(44,131,48,135)(50,121,54,125)(52,123,56,127)(57,63,61,59)(65,71,69,67)(66,84,70,88)(68,86,72,82)(73,155,77,159)(74,118,78,114)(75,157,79,153)(76,120,80,116)(81,87,85,83)(90,140,94,144)(92,142,96,138)(106,128,110,124)(108,122,112,126)(113,145,117,149)(115,147,119,151)(130,139,134,143)(132,141,136,137)(146,156,150,160)(148,158,152,154), (1,147,132,143,113)(2,144,148,114,133)(3,115,137,134,149)(4,135,116,150,138)(5,151,136,139,117)(6,140,152,118,129)(7,119,141,130,145)(8,131,120,146,142)(9,27,81,111,39)(10,112,28,40,82)(11,33,105,83,29)(12,84,34,30,106)(13,31,85,107,35)(14,108,32,36,86)(15,37,109,87,25)(16,88,38,26,110)(17,55,128,103,70)(18,104,56,71,121)(19,72,97,122,49)(20,123,65,50,98)(21,51,124,99,66)(22,100,52,67,125)(23,68,101,126,53)(24,127,69,54,102)(41,89,73,63,157)(42,64,90,158,74)(43,159,57,75,91)(44,76,160,92,58)(45,93,77,59,153)(46,60,94,154,78)(47,155,61,79,95)(48,80,156,96,62), (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( (1,83,57,71)(2,84,58,72)(3,85,59,65)(4,86,60,66)(5,87,61,67)(6,88,62,68)(7,81,63,69)(8,82,64,70)(9,89,24,130)(10,90,17,131)(11,91,18,132)(12,92,19,133)(13,93,20,134)(14,94,21,135)(15,95,22,136)(16,96,23,129)(25,79,125,151)(26,80,126,152)(27,73,127,145)(28,74,128,146)(29,75,121,147)(30,76,122,148)(31,77,123,149)(32,78,124,150)(33,43,104,143)(34,44,97,144)(35,45,98,137)(36,46,99,138)(37,47,100,139)(38,48,101,140)(39,41,102,141)(40,42,103,142)(49,114,106,160)(50,115,107,153)(51,116,108,154)(52,117,109,155)(53,118,110,156)(54,119,111,157)(55,120,112,158)(56,113,105,159), (1,7,5,3)(2,64,6,60)(4,58,8,62)(9,37,13,33)(10,101,14,97)(11,39,15,35)(12,103,16,99)(17,38,21,34)(18,102,22,98)(19,40,23,36)(20,104,24,100)(25,107,29,111)(26,51,30,55)(27,109,31,105)(28,53,32,49)(41,95,45,91)(42,129,46,133)(43,89,47,93)(44,131,48,135)(50,121,54,125)(52,123,56,127)(57,63,61,59)(65,71,69,67)(66,84,70,88)(68,86,72,82)(73,155,77,159)(74,118,78,114)(75,157,79,153)(76,120,80,116)(81,87,85,83)(90,140,94,144)(92,142,96,138)(106,128,110,124)(108,122,112,126)(113,145,117,149)(115,147,119,151)(130,139,134,143)(132,141,136,137)(146,156,150,160)(148,158,152,154), (1,147,132,143,113)(2,144,148,114,133)(3,115,137,134,149)(4,135,116,150,138)(5,151,136,139,117)(6,140,152,118,129)(7,119,141,130,145)(8,131,120,146,142)(9,27,81,111,39)(10,112,28,40,82)(11,33,105,83,29)(12,84,34,30,106)(13,31,85,107,35)(14,108,32,36,86)(15,37,109,87,25)(16,88,38,26,110)(17,55,128,103,70)(18,104,56,71,121)(19,72,97,122,49)(20,123,65,50,98)(21,51,124,99,66)(22,100,52,67,125)(23,68,101,126,53)(24,127,69,54,102)(41,89,73,63,157)(42,64,90,158,74)(43,159,57,75,91)(44,76,160,92,58)(45,93,77,59,153)(46,60,94,154,78)(47,155,61,79,95)(48,80,156,96,62), (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([(1,83,57,71),(2,84,58,72),(3,85,59,65),(4,86,60,66),(5,87,61,67),(6,88,62,68),(7,81,63,69),(8,82,64,70),(9,89,24,130),(10,90,17,131),(11,91,18,132),(12,92,19,133),(13,93,20,134),(14,94,21,135),(15,95,22,136),(16,96,23,129),(25,79,125,151),(26,80,126,152),(27,73,127,145),(28,74,128,146),(29,75,121,147),(30,76,122,148),(31,77,123,149),(32,78,124,150),(33,43,104,143),(34,44,97,144),(35,45,98,137),(36,46,99,138),(37,47,100,139),(38,48,101,140),(39,41,102,141),(40,42,103,142),(49,114,106,160),(50,115,107,153),(51,116,108,154),(52,117,109,155),(53,118,110,156),(54,119,111,157),(55,120,112,158),(56,113,105,159)], [(1,7,5,3),(2,64,6,60),(4,58,8,62),(9,37,13,33),(10,101,14,97),(11,39,15,35),(12,103,16,99),(17,38,21,34),(18,102,22,98),(19,40,23,36),(20,104,24,100),(25,107,29,111),(26,51,30,55),(27,109,31,105),(28,53,32,49),(41,95,45,91),(42,129,46,133),(43,89,47,93),(44,131,48,135),(50,121,54,125),(52,123,56,127),(57,63,61,59),(65,71,69,67),(66,84,70,88),(68,86,72,82),(73,155,77,159),(74,118,78,114),(75,157,79,153),(76,120,80,116),(81,87,85,83),(90,140,94,144),(92,142,96,138),(106,128,110,124),(108,122,112,126),(113,145,117,149),(115,147,119,151),(130,139,134,143),(132,141,136,137),(146,156,150,160),(148,158,152,154)], [(1,147,132,143,113),(2,144,148,114,133),(3,115,137,134,149),(4,135,116,150,138),(5,151,136,139,117),(6,140,152,118,129),(7,119,141,130,145),(8,131,120,146,142),(9,27,81,111,39),(10,112,28,40,82),(11,33,105,83,29),(12,84,34,30,106),(13,31,85,107,35),(14,108,32,36,86),(15,37,109,87,25),(16,88,38,26,110),(17,55,128,103,70),(18,104,56,71,121),(19,72,97,122,49),(20,123,65,50,98),(21,51,124,99,66),(22,100,52,67,125),(23,68,101,126,53),(24,127,69,54,102),(41,89,73,63,157),(42,64,90,158,74),(43,159,57,75,91),(44,76,160,92,58),(45,93,77,59,153),(46,60,94,154,78),(47,155,61,79,95),(48,80,156,96,62)], [(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)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 24 | 26 |
| 0 | 0 | 28 | 30 | 7 | 30 |
| 0 | 0 | 11 | 34 | 11 | 13 |
| 0 | 0 | 15 | 17 | 37 | 37 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,4,28,11,15,0,0,4,30,34,17,0,0,24,7,11,37,0,0,26,30,13,37] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 5 | 8A | ··· | 8P | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | M4(2) | C4○D4 | F5 | C2×F5 | D5⋊C8 | D5⋊M4(2) | D10.C23 |
| kernel | C42.6F5 | C4×C5⋊C8 | D10⋊C8 | Dic5⋊C8 | D5×C42 | C4×Dic5 | C4×C20 | C2×C4×D5 | C4×D5 | Dic5 | Dic5 | C42 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 16 | 4 | 4 | 1 | 3 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._6F_5
% in TeX
G:=Group("C4^2.6F5"); // GroupNames label
G:=SmallGroup(320,1016);
// by ID
G=gap.SmallGroup(320,1016);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=b^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
○
≀