metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.7F5, D10.10M4(2), Dic5.11M4(2), (C4×C20).4C4, Dic5⋊C8⋊5C2, D10⋊C8.5C2, C5⋊2(C42.6C4), (C4×Dic5).28C4, (D5×C42).16C2, C10.9(C2×M4(2)), C10.C42⋊6C2, C10.4(C42⋊C2), Dic5.26(C4○D4), C22.65(C22×F5), C2.10(D5⋊M4(2)), (C4×Dic5).323C22, (C2×Dic5).320C23, C2.9(D10.C23), (C2×C4×D5).31C4, (C2×C5⋊C8).4C22, (C2×C4).100(C2×F5), (C2×C20).101(C2×C4), (C2×C4×D5).359C22, (C2×C10).22(C22×C4), (C2×Dic5).170(C2×C4), (C22×D5).122(C2×C4), SmallGroup(320,1022)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 110 conjugacy classes, 46 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], 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 [×4], Dic5, C20 [×3], 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], C10.C42 [×2], D10⋊C8 [×2], Dic5⋊C8 [×2], D5×C42, C42.7F5
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, C22×F5, D5⋊M4(2) [×2], D10.C23, C42.7F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2b2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c3 >
►On 160 points
Generators in S160
(1 69 61 85)(2 82 62 66)(3 71 63 87)(4 84 64 68)(5 65 57 81)(6 86 58 70)(7 67 59 83)(8 88 60 72)(9 139 20 43)(10 48 21 144)(11 141 22 45)(12 42 23 138)(13 143 24 47)(14 44 17 140)(15 137 18 41)(16 46 19 142)(25 115 121 157)(26 154 122 120)(27 117 123 159)(28 156 124 114)(29 119 125 153)(30 158 126 116)(31 113 127 155)(32 160 128 118)(33 130 100 93)(34 90 101 135)(35 132 102 95)(36 92 103 129)(37 134 104 89)(38 94 97 131)(39 136 98 91)(40 96 99 133)(49 146 110 78)(50 75 111 151)(51 148 112 80)(52 77 105 145)(53 150 106 74)(54 79 107 147)(55 152 108 76)(56 73 109 149)
(1 83 57 71)(2 88 58 68)(3 85 59 65)(4 82 60 70)(5 87 61 67)(6 84 62 72)(7 81 63 69)(8 86 64 66)(9 89 24 130)(10 94 17 135)(11 91 18 132)(12 96 19 129)(13 93 20 134)(14 90 21 131)(15 95 22 136)(16 92 23 133)(25 79 125 151)(26 76 126 148)(27 73 127 145)(28 78 128 150)(29 75 121 147)(30 80 122 152)(31 77 123 149)(32 74 124 146)(33 43 104 143)(34 48 97 140)(35 45 98 137)(36 42 99 142)(37 47 100 139)(38 44 101 144)(39 41 102 141)(40 46 103 138)(49 118 106 156)(50 115 107 153)(51 120 108 158)(52 117 109 155)(53 114 110 160)(54 119 111 157)(55 116 112 154)(56 113 105 159)
(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)| (1,69,61,85)(2,82,62,66)(3,71,63,87)(4,84,64,68)(5,65,57,81)(6,86,58,70)(7,67,59,83)(8,88,60,72)(9,139,20,43)(10,48,21,144)(11,141,22,45)(12,42,23,138)(13,143,24,47)(14,44,17,140)(15,137,18,41)(16,46,19,142)(25,115,121,157)(26,154,122,120)(27,117,123,159)(28,156,124,114)(29,119,125,153)(30,158,126,116)(31,113,127,155)(32,160,128,118)(33,130,100,93)(34,90,101,135)(35,132,102,95)(36,92,103,129)(37,134,104,89)(38,94,97,131)(39,136,98,91)(40,96,99,133)(49,146,110,78)(50,75,111,151)(51,148,112,80)(52,77,105,145)(53,150,106,74)(54,79,107,147)(55,152,108,76)(56,73,109,149), (1,83,57,71)(2,88,58,68)(3,85,59,65)(4,82,60,70)(5,87,61,67)(6,84,62,72)(7,81,63,69)(8,86,64,66)(9,89,24,130)(10,94,17,135)(11,91,18,132)(12,96,19,129)(13,93,20,134)(14,90,21,131)(15,95,22,136)(16,92,23,133)(25,79,125,151)(26,76,126,148)(27,73,127,145)(28,78,128,150)(29,75,121,147)(30,80,122,152)(31,77,123,149)(32,74,124,146)(33,43,104,143)(34,48,97,140)(35,45,98,137)(36,42,99,142)(37,47,100,139)(38,44,101,144)(39,41,102,141)(40,46,103,138)(49,118,106,156)(50,115,107,153)(51,120,108,158)(52,117,109,155)(53,114,110,160)(54,119,111,157)(55,116,112,154)(56,113,105,159), (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( (1,69,61,85)(2,82,62,66)(3,71,63,87)(4,84,64,68)(5,65,57,81)(6,86,58,70)(7,67,59,83)(8,88,60,72)(9,139,20,43)(10,48,21,144)(11,141,22,45)(12,42,23,138)(13,143,24,47)(14,44,17,140)(15,137,18,41)(16,46,19,142)(25,115,121,157)(26,154,122,120)(27,117,123,159)(28,156,124,114)(29,119,125,153)(30,158,126,116)(31,113,127,155)(32,160,128,118)(33,130,100,93)(34,90,101,135)(35,132,102,95)(36,92,103,129)(37,134,104,89)(38,94,97,131)(39,136,98,91)(40,96,99,133)(49,146,110,78)(50,75,111,151)(51,148,112,80)(52,77,105,145)(53,150,106,74)(54,79,107,147)(55,152,108,76)(56,73,109,149), (1,83,57,71)(2,88,58,68)(3,85,59,65)(4,82,60,70)(5,87,61,67)(6,84,62,72)(7,81,63,69)(8,86,64,66)(9,89,24,130)(10,94,17,135)(11,91,18,132)(12,96,19,129)(13,93,20,134)(14,90,21,131)(15,95,22,136)(16,92,23,133)(25,79,125,151)(26,76,126,148)(27,73,127,145)(28,78,128,150)(29,75,121,147)(30,80,122,152)(31,77,123,149)(32,74,124,146)(33,43,104,143)(34,48,97,140)(35,45,98,137)(36,42,99,142)(37,47,100,139)(38,44,101,144)(39,41,102,141)(40,46,103,138)(49,118,106,156)(50,115,107,153)(51,120,108,158)(52,117,109,155)(53,114,110,160)(54,119,111,157)(55,116,112,154)(56,113,105,159), (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([(1,69,61,85),(2,82,62,66),(3,71,63,87),(4,84,64,68),(5,65,57,81),(6,86,58,70),(7,67,59,83),(8,88,60,72),(9,139,20,43),(10,48,21,144),(11,141,22,45),(12,42,23,138),(13,143,24,47),(14,44,17,140),(15,137,18,41),(16,46,19,142),(25,115,121,157),(26,154,122,120),(27,117,123,159),(28,156,124,114),(29,119,125,153),(30,158,126,116),(31,113,127,155),(32,160,128,118),(33,130,100,93),(34,90,101,135),(35,132,102,95),(36,92,103,129),(37,134,104,89),(38,94,97,131),(39,136,98,91),(40,96,99,133),(49,146,110,78),(50,75,111,151),(51,148,112,80),(52,77,105,145),(53,150,106,74),(54,79,107,147),(55,152,108,76),(56,73,109,149)], [(1,83,57,71),(2,88,58,68),(3,85,59,65),(4,82,60,70),(5,87,61,67),(6,84,62,72),(7,81,63,69),(8,86,64,66),(9,89,24,130),(10,94,17,135),(11,91,18,132),(12,96,19,129),(13,93,20,134),(14,90,21,131),(15,95,22,136),(16,92,23,133),(25,79,125,151),(26,76,126,148),(27,73,127,145),(28,78,128,150),(29,75,121,147),(30,80,122,152),(31,77,123,149),(32,74,124,146),(33,43,104,143),(34,48,97,140),(35,45,98,137),(36,42,99,142),(37,47,100,139),(38,44,101,144),(39,41,102,141),(40,46,103,138),(49,118,106,156),(50,115,107,153),(51,120,108,158),(52,117,109,155),(53,114,110,160),(54,119,111,157),(55,116,112,154),(56,113,105,159)], [(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)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 14 | 0 | 27 |
| 0 | 0 | 0 | 7 | 14 | 27 |
| 0 | 0 | 27 | 14 | 7 | 0 |
| 0 | 0 | 27 | 0 | 14 | 34 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 39 | 9 | 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 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 5 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 20 | 24 | 7 |
| 0 | 0 | 7 | 27 | 31 | 31 |
| 0 | 0 | 14 | 10 | 10 | 14 |
| 0 | 0 | 34 | 34 | 17 | 21 |
G:=sub<GL(6,GF(41))| [1,23,0,0,0,0,0,40,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[32,39,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[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],[8,5,0,0,0,0,10,33,0,0,0,0,0,0,24,7,14,34,0,0,20,27,10,34,0,0,24,31,10,17,0,0,7,31,14,21] >;
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 |
| type | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | C4○D4 | M4(2) | F5 | C2×F5 | D5⋊M4(2) | D10.C23 |
| kernel | C42.7F5 | C10.C42 | D10⋊C8 | Dic5⋊C8 | D5×C42 | C4×Dic5 | C4×C20 | C2×C4×D5 | Dic5 | Dic5 | D10 | C42 | C2×C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 3 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._7F_5
% in TeX
G:=Group("C4^2.7F5"); // GroupNames label
G:=SmallGroup(320,1022);
// by ID
G=gap.SmallGroup(320,1022);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,344,758,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2*b^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^-1,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀